# Full text of "A Course Of Differential Geometry"

## See other formats

TEXT FLY WITHIN THE BOOK ONLY 00 OU1 66578 >m UNIVERSITY LIBRARY Call No. ' 'C Accession ~~ fflh book should be retured on or before the date I& ijKlfEed below, A COURSE OF DIFFERENTIAL GEOMETEY Oxford University Press London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calcutta Madras Shanghai Humphrey Milford Publisher to the UNIVERSITY A COURSE OF DIFFERENTIAL GEOMETRY^ BY THE LATE JOHN EDWARD CAJMPJBELL M.A. (OXON.), HON.D.SC, (BELFAST), F.R.S. FELLOW OF HERTFORD COLLEGE, OXFORD PREPARED FOR THE PRESS WITH THE ASSISTANCE OF E, B. ELLIOTT M.A., F.R.S., EMERITUS PROFESSOR OXFORD AT THE CLARENDON PRESS 1926 Printed in England At the OXFORD UNIVERSITY PRESS By John Johnson Printer to the University PREFACE MY father .had spent most of his spare time since the War in writing this book. Only two .months before his death, while on our summer holiday in 1924, he had brought some of the chapters with him, and sent off the final draft of them to the Clarendon Press. Even on these holidays, which he greatly enjoyed, we were all accustomed to a good deal of work, and it was an unexpected pleasure to find that with these once dispatched to the press he took an unusually complete holiday. While rejoicing that he was so far able to com- plete the book, we are sorry that a last chapter or appendix in which he was greatly interested was hardly begun. Apparently this was to deal with the connexion between the rest of the book and Einstein's theory. To the mathematical world his interest in this was shown by his Presidential address to the London Mathematical Society in 1920 to his friends by the delight he took on his frequent walks in trying to explain in lucid language something of what Einstein's theory meant. vi PREFACE We cannot be too grateful to Professor Elliott, F.R.S., s,n old friend of many years standing, for preparing the book for the press and reading and correcting the proofs. No labour has been too great for him to make the book as nearly as possible what it would have been. And the task has been no light one. We should like to thank the Clarendon Press for their unfailing courtesy and for the manner in which the book has been produced. J. M. H. C. Christmas 1925. EDITOR'S NOTE MY dear friend the author of this book has devoted to preparation for it years of patient study and inde- pendent thought. Now that he has passed away, it has been a labour of love to me to do my best for him in seeing it through the press. As I had made no special study of Differential Geometry beforehand, and was entirely without expertness in the methods of which Mr. Campbell had been leading us to realize the importance, there was no danger of my converting the treatise into one partly my own. It stands the work of a writer of marked individuality, with rather unusual instincts as to naturalness in presentation. A master's hand is shown in the analysis. Before his death he had written out, and submitted to the Delegates of the University Press, nearly all that he meant to say. An appendix, bearing on the Physics of Einstein, was to have been added ; but only introductory statements on the subject have been found among his papers. Unfortunately finishing touches, to put the book itself in readiness for printing, had still to be given to it. The chapters were numbered viii EDITOR'S NOTE in an order which, rightly or wrongly, is in one place here departed from, but they stood almost as separate monographs, with only a very few references in general terms from one to another. To connect them as the author would have done in due course is beyond the power of another. The articles, however, have now been numbered, and headings have been given to them. Also some references have been introduced. The text has not been tampered with, except in details of expression ; but a few foot-notes in square brackets have been appended. E. B. E. TABLE OF CONTENTS CHAPTER I TENSOR THEORY PAGE The n-way differential quadratic form (1) 1 The distance element. Euclidean and curved spaces (2) 2 Vectors in a Euclidean space which trace out the space of a form (3) 4 Christoffel's two symbols of three indices (4) . v ~r*~~^ . . 6 Some important operators (5) \^^^ ^^"'^' Conclusions as to derivatives of ajj., a ll> , and i lg#~rt'(6) . ****T . 8 Tensors and tensor components defined (7j^X*C .... 9 The functions a^ and a lk aie tensor components (8) ,, . . . 10 Expressions for second derivatives when a^dx^dx^ = o! ^dx {dx ^ (9) 11 Tensor derivatives of tensor components are tensor components (10) 12 Rules and definitions of the tensor calculus (11) .... 14 Beltrami's three differential parameters (12) 16 Two associated vector spaces. Normals to surfaces (13) . . . 17 Euclidean coordinates at a point (14) 18 Two symbols of four indices which are tensor components (15) . 19 A four-index generator of tensor components from tensor com- ponents (16) 21 Systems of invariants (17) .22 An Einstein space, and its vanishing invariants (18) ... 23 CHAPTER II THE GROUND FORM WHEN n = 2 '' Alternative notations (19) 25 An example of applicable surfaces (20) 26 Spherical and pseudo-spherical surfaces. The tractrix revolution surface (21) 27 X TABLE OF CONTENTS PAGE Ruled and developable surfaces. The latter applicable on a plane (22) 28 Elliptic coordinates (23) 31 The invariant K (24) 32 Determination of a ^ such that A (<, t//>) = (25) .... 34 Reduction of a ground form when K is constant (26) . . . o 34 The case of A (JST) (27) 36 The case when A 2 K and A K are functions of K (28) ... 37 Conditions for equivalence in the general case (29) .... 38 The functions called rotation functions (30) 39 Integration of the complete system of 24 (31) . . . . 40 CHAPTER III GEODESICS IN TWO-WAY SPACE Differential equation of a geodesic ( 32) . ^S .... 42 Another form of the equation (33) . s . . , " .43 Condition that orthogonal trajectories be geodesic^ {34) . . 44 Geodesic curvature (35, 36) ". 45 Polar geodesic coordinates (37) . - 47 Recapitulation. Parallel curves (38) 49 Notes regarding geodesic curvature (39, 40) 51 Integration of geodesic equations when K\* constant (41) . . 54 Focal coordinates (42) 55 Explicit expressions for symbols [ikj] and for K (43) ... 55 Liouville's special form (44) 58 Null lines. Complex functions of position (45) . . . .58 Conjugate Harmonic Functions. Mapping on a plane (46) . . 60 CHAPTER IV TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE A quaternion notation (47) 62 Introduction of new fundamental magnitudes and equations (48) . 63 Connexion of the magnitudes with curvature (49) .... 64 The normal vector determinate when the functions Qy are known (50) 65 Reference to lines of curvature. The measure of curvature (51) . 68 Tangential equations. Minimal surfaces (52) 69 Weingarten or W surfaces (53) 71 TABLE OP CONTENTS XI PAGE An example of W surfaces (54) 72 The spherical and pseudo-spherical examples (55) . ..... 7S Reference to asymptotic lines (56) 75 Equations determining a surface (57) 78 The equation for the normal vector in tensor form (58) ... 79 Introduction of a new vector (59) 80 Orthogonally corresponding surfaces (60) 81 Recapitulation (61) 82 Relationship of surfaces z and f (62) 83 Association of two other surfaces with a ^-surface (63, 64) . . 84 CHAPTER V DEFORMATION OF A SURFACE, AND CONGRUENCES Continuous deformation of a surface (65) 86 A vector of rotation (66) 87 Geometrical relationship of surfaces traced out by certain vectors (67,68) 88 A group of operators, and a system of twelve associated surfaces traced out by vectors (69, 70) 89 The twelve surfaces form three classes of four (71) .... 91 A case in which one surface is minimal (72) 92 Congruences of straight lines (73) 93 Focal planes and focal points of a ray (74) 94 Limiting points. The Hamiltonian equation. Principal planes (75) 96 Principal surfaces, and the central surface (76) . . .97 The focal surface (77) ... 98 Rays touch both sheets of the focal surface. The congruence of rays of light (78) 98 Refraction of a congruence. Malus's theorem (79) .... 100 The Ribaucourian congruence (80) 101 The Isotropic congruence. Ribaucour's theorem (81) . . . 102 W congruences (82) 103 Congruence of normals to a surface (83) 104 Reference to lines of curvature (84) 105 Tangents to a system of geodesies (85) 105 Connexion of W congruences which are normal with If surfaces (86) 106 Surfaces applicable to surfaces of revolution, and W normal congruences (87) 107 .Surfaces of constant negative curvature (88) 108 Xll TABLE OF CONTENTS CHAPTER VI CURVES IN EUCLIDEAN SPACE AND ON A SURFACE. MOVING AXES PAGE ferret's formulae. Rotation functions ( 89) r 110 Codazzi's equations (90) Ill Expressions for curvature and torsion (91) 113 Determination of a curve from Serret's equations (92) . . .114 Associated Bertrand curves (93) 116 A curve on a surface in relation to that suiface (94) . . .117 Formulae for geodesic torsion and curvature (95) .... 120 Surfaces whose lines of curvature are plane curves (96j . . . 121 Enneper's theorem (97) 125 The method of moving axes (98) 125 Orthogonal surfaces (99) 127 CHAPTER VII THE RULED SURFACE Unit orthogonal vectors ( 100) 129 The ground form and fundamental magnitudes (101) . . . 130 Bonnet's theorem on applicable ruled surfaces (102) . . . 131 Ground forms applicable on a ruled surface (103) . . . .133 Case of applicability to a quadric (104) 135 Special ground forms. Binomials to a curve. Line of stric- tion(105) 136 Constancy of anharinonic latios. Applicable ruled surfaces and surfaces of Revolution (106) 138 Surfaces cutting at one angle all along a generator (107) . . 139 The ruled surfaces of an isotropic congruence (108) . . .141 CHAPTER VIII THE MINIMAL SURFACE Formulae and a characteristic propeity ( 109) .... 143 Reference to null lines. Stereographic projection (110) . . . 145 The vector of a null curve (111) 147 Self-conjugate null curves. They may be (1) unicursal, (2) al- gebraic (112) 147 TABLE OF CONTENTS Xlll FAGE Generation of minimal surfaces from null curves. Double minimal surfaces (113) t . 149 Henneberg's surface (114) 151 Lines of curvature and asymptotic lines on minimal surfaces (115) . 152 Associate and adjoint minimal surfaces (116) 153 CHAPTER IX THE PROBLEM OF PLATEAU AND CONFORMAL REPRESENTATION The minimal surface with a given closed boundary ( 117) . . 155 The notation of a linear differential equation of the second order with three singularities (118) 157 Conforinal representation on a triangular area (119) . . . 158 The w-plane or part of it covered with curvilinear triangles (120) . 161 Consideration of the case when triangles do not overlap (121) . 163 Case of a real orthogonal circle as natural boundary (122) . . 165 Fundamental spherical triangles when there is no natural boun- dary (123) 166 Summary of conclusions (124) 168 Representation of the o>plane on a given polygon (125, 126) . . 168 CHAPTER X ORTHOGONAL SURFACES A certain partial differential equation of the third order ( 127) . 172 A solution led to when functions satisfying a set of three equations are known (128) 174 The vector q(Xq~ l , where OK is a vector and q a quaternion (129) . 174 Passage from set to set of three orthogonal vectors (130) . . 175 Rotation functions (131) 177 A vector which traces out a triply orthogonal system (132) . . 178 Lines and measures of curvature (133) 179 Linear equations on whose solution depends that of the equation of the third order (134) 181 Synopsis of the general argument (135) 182 An alternative method indicated (136) 184 Three additional conditions which may be satisfied (137) . . 185 ^Orthogonal systems from which others follow by direct operations (138) 186 XIV TABLE OF CONTENTS CHAPTER XI INFERENTIAL GEOMETRY IN n-WAY SPACE PAGE Geodesies in n-way space ( 139) 188 Geodesic polar coordinates and Euclidean coordinates at a point (140) 190 Riemann's measure of curvature of n-way space (141) . . . 193 Further study of curvature. The Gaussian measures for geodesic surfaces. Orientation (142) - 194 A notation for oriented area (143) 198 A system of geodesies normal to one surface are normal to a system of surfaces (144) 199 The determination ot surfaces orthogonal to geodesies and of geodesies orthogonal to surfaces (145) 203 A useful reference in (n + 1)- way space (146) 205 Geometry of the functions fi t -fc (147) 207 The sum of the products of two principal curvatures at a point (148) 209 Einstein space (149) 211 An (n+ l)-way Einstein space surrounds any given n-way space (150-4) 212 CHAPTER XII THE GENERATION OF AN (w + l)-WAY STATIONARY EINSTEIN SPACE FROM AN n-WAY SPACE Conditions that the (n + l)-way Einstein space surrounding a given n-way space be stationary ( 155) 220 Infinitesimal generation of the (n-f-l)-way from the n-\\ay form (156) 222 Restatement and interpretation of results (157) .... 225 A particular case examined when n = 2 (158) 226 General procedure in looking for a four- way stationary Einstein space (159) 228 Conclusions as to curvature (160) ....... 229 CHAPTER XIII n-WAY SPACE OF CONSTANT CURVATURE Ground form for a space of zero Riemann curvature ( 161) . . 231 Ground form for a space of constant curvature for all orientations (162) 232 TABLE OF CONTENTS XV PAGE Different forms for these spaces (163) 234 Geodesic geometry for a space of curvature +1 (164) . ., . 236 Geodesies as circles (165) . .' 237 Geodesic distance between two points (166) 238 Coordinates analogous to polar coordinates (167) .... 239 The three-way space of curvature 4- 1 (168) 240 The geometry of the space (169) 241 Formulae for lines in the space, and an invariant (170) . . . 243 Volume in the space (171) 246 An n-way space of constant curvature as a section of an extended Einstein space (172) 247 CHAPTER XIV n-WAY SPACE AS A LOCUS IN (w + l)-WAY SPACE A space by which any n-way space may be surrounded ( 173) . 250 Curvature properties of this surrounding space (174, 175) . .251 A condition that the surrounding space may be Euclidean (176) . 253 Procedure for applying the condition when n >2 (177) . . . 255 The n-way space as a surface in the Euclidean space when this exists (178, 179) 255 INDEX 259 CHAPTER I TENSOR THEORY 1. The n-wa,y differential quadratic form. Let us consider the expression a ik dxidx k , \-= 1 ...n (1.1) which is briefly written for the sum of r^ such terms, obtained by giving to i, k independently the values 1, 2, ... n. If, for instance, n = 2, the expression is a short way of writing a n dxl + 2ct l2 dj' l dx 2 + a 2 ^dx^ ; for we are assuming that tfc=fci- (*- 2 ) Let us also denote by a ih the result of dividing by a itself (\) i+1 * times the determinant obtained by erasing the row and the column \vhi cli contain a in the determinant a = (1.3) The coefficients a ik ... are at present arbitrarily assigned functions of the variables x l ...^' n> limited only by the condition that a is not zero. When we are given the coefficients a ik as functions of their arguments, there must exist r functions X l ...X r , r = -in(?M-l), of the variables a^ ... x n , such that dXl + ...+ dXl = a ik dx 4 dx k . (1.4) The differential equations which will determine these functions are %g ^% = a ik . (1.5) *i ^ x k Just as in the expression a ik dx t dx k the law of the notation is that, whenever a suffix, which occurs in one factor of TENSOR THEORY a product, is repeated in another factor^ the sum of all such products is to be taken, so here the above differential equation is the short way of writing As there are just as many unknown functions as there are differential equations to be satisfied, we know that the functions X l ... X r must exist. The actual solution of this system of differential equations is, however, quite another matter, and questions connected with the solution form a chief part in the study of Differential Geometry. 2. The distance element. Euclidean and curved spaces. If we regard x lt ..x n as the coordinates of a point in an 7i-way space, then, X 1 ... X r being functions of y\ ...x n . we may regard this space as a locus in r-way Euclidean space ; and we may regard dk as the distance between two neighbour- ing points x l . . . x n an d x 1 + dx l ...x n + dx n , where ds is defined by ds z = (f{k d Xi dx k . (2.1) Thus, if n = 2, the two-way space given by ds 2 = atffdxidxfr lies within our ordinary Euclidean space, and it is with this space that Differential Geometry has hitherto been chiefly concerned. If 7i = 3, the * curved * three-way space lies, in general, within a Euclidean six-way space. If, however, the coefficients a^, instead of being arbitrarily assigned functions of their arguments x lt o/ 2 , # 3 , satisfy certain conditions, the Euclidean space may be only a five- way space, or even only a four- way space. In yet more special cases the three-way space may not be * curved 1 at all, but only ordinary Euclidean space with a different coordinate system of reference. If n = 4, the curved four-way space lies, in general, within a Euclidean ten- way space, and so on. We know what a curved two-way space within a Euclidean three-way space means, being a surface : but what does a ' curved ' three-way space mean ? We have not, and we DISTANCE ELEMENT. EUCLIDEAN AND CURVED SPACES 3 cannot have, a conception of a four- way space, Euclidean or otherwise, within which the three-way space is to be curved. But by thinking of the geometry associated with the form cfe 2 = a u dxl + 2<f 12 davir 2 + ff 22 <fo5 ( 2 2 ) we say that it is that of a curved two-way space ; and we know that it is, in general, different from the flat Euclidean plane geometry associated with the form d*> = dxl+dxl. (2.3) We can distinguish these two geometries without any leference to the Euclidean three-way space, or any other three-way space. This distinction we, with our knowledge of a three-way Euclidean space, characterize by saying that the first space is curved and the second flat, or Euclidean. This is what we mean when we say that the space given by ds* = (tfadxidxje (2 4) is, in genera], a curved space, whilst that given by d# = <Y.Y? + ...+<W:* (2.5) is a flat space. We shall find that a geometrical property will be associated with a curved space, which will distinguish it from a flat space. If we have no real knowledge of a space of more than three dimensions, we have at least no knowledge that it does not exist: and, by analogy from our knowledge both of a two-way space and a three-way space, we are able to make use of the ideas of higher space to express analytical results in an interesting form. The space in which wo live may, or may not, be flat or Euclidean. Up till quite recently it has been assumed to be flat, and the geometry which has been built up has been that associated with the form The geometry which we wish to know about to-day would be that associated with the form ds 2 = afodxtdxfa \ = 1 ... 4, where x...x are functions of the three variables which 4 TENSOR THEORY locate an event in space, and a fourth variable which locates it in time. The geometry of Euclidean space is much simpler than the geometry associated with the more general form, and its properties have been more studied. It may therefore be of advantage, at least in some ways, to regard the form ds* = aftdxidxje (2.1) as that of an ft- way locus in a tiat r-way space, although r is generally a much larger number than n. 3. Vectors in a Euclidean space which trace out the space of a form. Let i' \ i" , i'" ... be r unit vectors in the Euclidean space and let y and z be vectors given by What we call the scalar product of the two ^vectors y and z is denoted by yz and defined by y^ + y' Z ' + y"z"+... =0. (3.2) The cosine of the angle between the vectors is defined as -yz and may be written _^_ . (3 . 3) \yy zz We shall generally write yy as 2/ 2 , but we must remember then that the root of y 1 is not y. The numbers y', y" ... are called the components of the vector y : they are ordinary scalar numbers. Now let z be a vector whose components are functions of the n parameters x l ... x n . Denoting the derivative of z with respect to x r by z r , we have dz = z p dx p in the notation we have explained, which is the foundation of the Tensor Calculus. We therefore have dzdz = ZiZkdXidXfr. (3.4) VECTORS IN A EUCLIDEAN SPACE 5 The vector z traces out an n-w&y space within the Euclidean r-way space, and in this ?i-way space the element of ' length ' is given by efc = -cfe<fe; (3.5) and therefore, if we take a ik = ~Wk> (3.6) we have cfes We say that cite is an element in this space, and we notice that an element has direction as well as length. The element is localized at the extremity of the vector z ; the element lies in the w-way space, but the vector lies in the r-way Euclidean space. The direction cosines of the element in the r-way space are , dx n dx p **-*' **-*"- (3 * 7) We write fp = /> s 7 > els and we speak of J , 2 , ... g as the direction cosines of the element in the ?i-way space given by, or associated with, ds 2 = a The upper affixes in g l ... * have, of course, no implication of powers as in ordinary algebra. The notation introduced is in accordance with that of the tensor calculus which we are leading up to. In accordance with that calculus we ought to write the variables x l . . . x n as a; 1 ... x n , but we do not do so, as the notation x l .. f x )l is at present too firmly fixed perhaps. If CD is the angle between two elements, drawn through the extremity of 0, whose direction cosines with respect to the n- way space are i 2 tn respectively, q = u pq l '*] (1 - (3.8) 6 TENSOR THEORY It should be noticed that ^pq^rj^ means precisely the same thing as fl^^V' Repeated suffixes are called dummy suffixes and'can be replaced by any other dummy suffixes. The chief rule that we need to follow is not to use the same dummy more than twice in an expression containing a number of factors. * It should be noticed that the angle, for which we have found an expression, is that between two elements drawn through the same point, viz. the same extremity of the vector z. We have no expression for the angle between two elements at different points in our 7i-way space. This is something that distinguishes the geometry connected with the form ds 2 a^dx^lx^ from the geometry of Euclidean space. 4. ChristofTel's two symbols of three indices. Let . (4.1) This is the definition of Chris toffel's three-index symbol of the first kind. It is exceedingly important in the theory of differential geometry. The first two suffixes are inter- changeable. We may write it sometimes in the form T t when we regard i and k as fixed suffixes. Since fl<fe = -^i we see that (ikl) = ~^^, (4 2 ) , Wz where z ijf = .- v ll{ dx.^rfr We introduce the symbol e\ to denote zero if i and k are unequal and unity if i and k are equal. We do not write J as equal to unity, for by our convention e{=j+e!| + ... + = n. In employing dummy suffixes it is best to employ a letter to which we have not attached a definite connotation. From the property of determinants and their first minors we see that a it 0kf = ^ m (i . 3) CHRISTOFFEL S TWO SYMBOLS OF THREE INDICES Let {ikj} =o#(arf); (4.4) then {ikj} is Christoffel's three-index symbol of the second kind. The first two suffixes are interchangeable and it may be written TJ when we regard i and k as fixed suffixes. We have at once (/&j) ^ ajt {ikt}. (4 . 5) 5. Some important operators. Even already we have come across a number of functions of the vaiiables which we denote by integers attached to a certain letter. Thus we have the fundamental functions denoted by u^...; we have the direction cosines denoted by 1 ... n , and the functions a ik .... More generally we may have a number of functions of the variables, say 6, <, >//, ... and we may form a function of 6, 0, ^Jr, ... and their derivatives with respect to the variables. It may be that the function thus arrived at may bs denoted by where a, j8, ... are integers of the upper row, upper integers we call them, and a, b y ... are lower integers. These integers may take independently any of the values 1, 2, ... u and thus indicate how the function 7', "" is formed. The number -/ a, 5, ... of the upper integers is not necessarily equal to the number of the lower integers. It may be that there are no integers in the upper row, or none in the lower, or even none in either. We shall come across many functions which may be ex- pressed in this manner, and we have come across some. In connexion with functions which are expressed in the above form there are n operators which are of fundamental importance in tensor theory. These operators may be written 1,2, 3,...rc, where p denotes the operator and where (') denotes the operation of substituting t for A, A. TENSOR THEORY A being any upper integer, and where (*) has a similar meaning with respect to a lower integer. (5.3) the which occurs on the right is a dummy suffix, and thus, for instance, We notice that the definite integers 1, 2, ... ?i are not dummies, and we should avoid the use of 71 as a dummy. -x - - /e e ^ We write 1 a- P =l> 1 a ( 5 5 ) By aid of the symbolism thus introduced we can avoid a prolixity which would otherwise almost bar progress. A very little practice will enable one to use this symbolism freely, and when necessary to express the results explicitly. 6. Conclusions as to derivatives of a ijc) a* 1 *, and We see from the definition that ; (6.1) and therefore the operator p annihilates each of the functions a ik , which, of course, could have been written T ik . We have a a ^ = ; and therefore /fc r a* 7 + a u (tpk) + a u (kpt) = 0. dx p It follows that a**a tk <- (fi + a** {tpq}+aH {kpi} = ; that is, a ty +a < {pg} +a {ipi} =0. (6 . CONCLUSIONS AS TO DERIVATIVES 9 It follows that the operator p also annihilates each of the functions a ik . By the rule for the differentiation of a determinant = a (tl )( l (ptq) + a a!"l (qtp) - 2a{i>tp], <* i i or - </2 a*{ptp}. (6.3) This formula will be required later. [It should be re- membered that the symbol on the right stands for the sum of n symbols, with p 1, 2, ... a. ] 7. Tensors and tensor components defined. We must now explain what is meant by a tensor. We have seen how functions denoted by r a > Py (iy 6. * * * may be derived from functions $,</>, ty ... and their derivatives with respect to X 1 ... x n . The different functions obtained by allowing the integers to take all values from 1 up to n are called components of the set. Suppose that we transform to ne\v variables a;'. ...x' n , and that & denotes the expression of 6 in terms of the new variables, and that (/>', ty' ... have similar meanings. Suppose further that r/ya',0', / / i./ M (i f) ... * * are functions formed from 0', X , ^', ... and their derivatives with respect to the new variables x\ ...x }l by exactly the same rules as the functions rot, A... a, b, ...'' were formed from 0, 0, \/r, and their derivatives with respect 10 TENSOR THEORY We say that 7^6,' '!.' are components of a tensor if TV ',', ... __ jta^ ^ ^V^V nr*,0,... * ;*,... -5^,a*v" ^ >V" ft ' 6 ""' Notice that the integers on the left are not dummies but that the integers a, )8, ... a } 6, ... on the right are, Notice also that the above equation must hold for all values of the integers on the left if the expressions r ,*,... a, b, ... are to bo tensor components. This is the formal definition : we shall immediately come across examples of ten&ors which will illustrate the definition. 8. The functions a^ and a ijc are tensor components. If we transform to new variables x\ ... x' n , the expression for the square of the element of length must remain unaltered in magnitude though its form may change We therefore have a pq dx p dx q = "\ndx\dx' p and so = " <'> Thus the functions a ik ... satisfy the condition for being tensor components. Again from the fundamental equality , we have a na ;/ = a Notice that q and X are no longer dummy suffixes in this <>x equality, Multiply across by a' KS ^~ , then we have 8 P t -r = a > ' FUNCTIONS a ik AND a lk ARE TENSOR COMPONENTS 11 The'exprossion on the right hand of this equality is **V ^ _ ^s ^r _ ^V _ r _ -p . . ' ~~ ' ~~ ~ ~ M ' therefore a lia (a r l> - a'** ^f r-^ 2 ) = 0. jq \ <)xl)x/ and This equation holds for all values of p, </, and r, and there- fore, as the determinant a is not zero, wo must have It follows that the functions a* k ... also satisfy the condition of being tensor components. 9. Expressions for second derivatives when We have z' r = where z f is the expression of z in terms of the new variables, and %' denotes ^ and z' a denotes ^ , ~ . . It follows, ? Zx' r W * p <>Mq since by (4 . 2) (pqr) 1 = -^2%, that Notice, that we see, from this equation, that Chris toffel's three-index symbols of the first kind do not satisfy the con- dition of being tensor components. Multiply across by af r * ^ * , and we have B - n -a _^- 7 - J> ^ ___ x4 ___ "But, by (8. 2), a' = a'* 12 TENSOR THEORY and therefore the right-hand member of this equation becomes We therefore have the fundamental formula in the trans- formation theory Similarly we have ,, ^ da' a a; 7 . J. (9.4, 10. Tensor derivatives of tensor components are tensor components. We must now show that the operators 1,2, ...n, when applied to any tensor components, generate other tensor components. Let MS L.... X'... and assume that jT** ^ '" ... are tensor components. We have >jv >*',.. = T*'^~' MN JL a', b',... JL a, b, ... " IV which we briefly write 2 T/ = TJUN. Expanding -j-M, using the for dx p' tion theory which have been obtained, Expanding -j-M, using the formulae of the transforma- and therefore (10.1) TENSOR DERIVATIVES OF TENSOR COMPONENTS 13 Similarly we have and therefore p' N = (^-/^}( l lt }-{^p't}'( l lt ))jf. (10.2) ^X ^\ Now T=--T i We have written x , a -rJ- . . . simply as M y but we must o T* Q 3* i.t note that Jlf has the upper integers a, 6, ... (as well as the lower integers a', &', ...) and that the upper integers in M are the same as the lower integers in T. Similarly we note that the lower integers in N are the upper integers in T. It follows that if we remember that these lower integers in N and upper integers in T are just dummies. We have similarly M{nql] (') T = T{tq\] (*)Jf, (10.5) M{w't}'fyT=T{V*}'(D*I> (^.7) and therefore $>' (TMN) = ^ MNqT. That is, p'T' = qT^~ ^~ ... r-A^-^ -^-' ..., (10.8) and therefore '7^ a ^'--- 1 t . are tensor components. J. Ji Q 0) , , . This is a very important theorem in the tensor calculus. It is the rule of taking what we call the tensor derivative 14 TENSOR THEORY and we see that the tensor derivative of a tensor component is a tensor component. We denote the p derivative by T::!;;:. P - 00.9) 11. Bules and definitions of the tensor calculus. We have now proved the most important theorem in the tensor calculus: its proof depended on the transformation theorems. These theorems, having served their purpose, disappear, as it were, from the calculus. There are some simple rules of the calculus which we now consider. The product of two tensors is a tensor whose components are the products of each component of the first and each component of the second tensor. The upper integers of the product are the upper integers of the two factors, and the lower integers of the product are the lower integers of the two factors. Two tensors of the same character that is, with the same number of each kind of integers, upper and lower can be added, if we take together the components which have the same integers. They can also be combined in other ways, as we shall see. We form the tensor derivative of the product of two tensors by the same rule as in ordinary differentiation. The tensors a^ and a ift are called fundamental tensors. We have seen that they have the property of being annihilated by any operator jp. As regards tensor derivation they there- lore play the part of constants. The symbol e^ satisfies the definition of a tensor. It also is called a fundamental tensor. Any tensor, formed by taking the product of a tensor and a fundamental tensor, is said to be an associate tensor of the tensor from which it is derived. Suppose that 2 a be '" * s an y tensor. The tensor itself is the entity made up of all its components, formed by allow- ing a, /?, y, ..., a, 6, c, ... to take all integral values from 1 up to n. Suppose now, that instead of taking all the RULES AND DEFINITIONS OF THE TENSOR CALCULUS 15 components, we take those in which one of the upper integers, say a, is equal to one of the lower integers, say^fc. The entity we thus arrive at will be a tensor. For The tensor thus arrived at is denoted by nrfp, 0,7. ... * ,P,c, ... 1 'A 4- u 7^,0,7,... and is said to be y a> &> c> contracted with respect to a, 6. We can contract a tensor with respect to any number of upper integers and an equal number of lower integers. ^l^r(X, If we take the tensor J[ a b ' an associate tensor would be a 7^, e a -/ a, 6 and we might write this y a ' & > and as it is contracted with respect to two upper integers and r^ & ^7"fft/3 ^Trt(X/3 So we may write a^tf / pg ns y We shall often use this contraction when we are consider- ing associate tensors The rank of a tensor is the number of integers, upper and lower, in any component. When the rank is zero the tensor is an invariant. When the rank is even we can form an associate tensor which will be an invariant. When the rank is odd we can form an associate tensor of rank unity. When the rank is unity the tensor may be said to be a vector in the 7i- way space : a contravariant vector if the integer is an upper one, a covariant vector if the integer is a lower one. But it must be carefully noticed that when we think of a- vector in the flat r-way space, we are thinking of the word vector in a different sense, Thua the vector z which traces 16 TENSOR THEORY out the /i- way space is not an invariant, but rather the entity of r invariants, and so as regards the derivatives of z. In the r-way space they are all vectors, hut the coefficients of the vectors i', i" ... come under the classification of tensors. If we bear this distinction in mind we shall not be misled, and we may gain an advantage by combining the two notions. It is a useless exaggeration of the great advantages of the tensor calculus to ignore the calculus of Quaternions. Wo certainly cannot afford to give up the aid of the directed vector notation in the differential geometry of flat space within which lies our vi-way curved space. 12. Beltrami's three differential parameters. If we take any function U of the variables, then u,,u t , ..u n will be tensor components. The tensor derivative of an invariant is just the ordinary derivative; 4ind therefore the above functions are just the same as U. l9 U. 99 ...U. n . (12.1) [For the notation see (5 . 5) and (10 . 9).] But if we take the second tensor derivatives we come across different functions from the ordinary second derivatives. These second tensor derivatives we denote by U . ik ... where 0-. tt =Z7- tt -{ifc}tf,. (12.2) These we have proved are tensor components ( 10), whereas the ordinary second derivatives U ik are not. It would be a useful exercise to prove that the functions U.^... are tensor components : it might make the general theorem, whose proof is rather complicated, more easily understood. The square of the tensor whose components are U l ... U n is a tensor whose components are U^U^ C . If we form the associate tensor a^U^U^ we have an invariant which is denoted by A ([/), so that &(U) =a ik U { U k . (12.3) This is Beltrami's first, differential parameter. Similarly by forming the tensor which is the product of BELTRAMIS THREE DIFFERENTIAL PARAMETERS 17 the two tensors whose components are t^ ... U n and V l ... V n , and taking the associate tensor a ik U i V k , we have B.eltrarni's * mixed* differential parameter A (17, V) = a ik U 4 V k . (12.4) We. also have Beltrami's second differential parameter & 2 (U) = a? k U. ik . (12.5) Clearly all these ' difFerential parameters ' as they are called are invariants. They are of great utility, as we shall find, in differential geometry. 13. Two associated vector spaces. Normals to surfaces. Returning now to the vector z, whose extremity traces out the n-way space within the flat r-way space, we have, see (5.2) and (12.2), *-M = *<ft-{'^}**- t 13 - 1 ) Clearly the components of this vector z . ^ are tensor com- ponents. We have ^/*,(>i+ 1) vectors z. ik and we have n vectors z i ; as these (%n(n+ 1) +n) vectors all lie in a %n(n+l) flat space there must be n linear equations connecting them. These vectors all depend on the parameters x l9 ..x nt and we may regard them as all localized at the extremity of the vector 0. Now, see 4, Z ' ih z p = z iJe Z p ~ tp { ikt } = 0. (13.2) We thus see that the vector z . i1c is perpendicular to every element in the ?i-way space drawn through the extremity of z. Let one of the n equations which connect the vectors *-ifc' *; bo b ik z -ik + b t-t= > where b ik ...b t ... are scalars. Multiply the equation by and take the scalar product : then, since we have that is, t } t a tp ~ 18 TENSOR THEORY and therefore, since the determinant a cannot be zero, we have 6j = 6 a = .. = 0. (13.3) It follows that the n linear equations connect the vectors z> ih ... only. At any point of the ?i-way space, therefore, there are n vectors z l ...z n generating a flat ii-way space; and there are -|/i(/i-f-l) vectors .^., only %n(nl) of which are linearly independent, and these generate a %n(nl) Hat space. These two flat spaces, associated with the point x l ... x n , are such that every element in the one space, drawn through the extremity of 0, is perpendicular to every element in the other space, drawn through the extremity of z. Thus when n is equal to ?, as it is in ordinary differential geometry, the vectors 3 . u , z . 12 , z . 22 (13.3) are parallel to the normal at the extremity of z which traces out the surface we are concerned with. 14. Euclidean coordinates at a point. Associated with every point x l ...x n we have a special sjstem of coordinates which we call the Euclidean coordinates of the point. They are very helpful in proving tensor identities, which without their aid would prove very laborious. At the point under consideration a ik ... (ikj) ... are constants. Let another set of constants be defined by a ik = b it b ki> b ik = 6 M' i 14 - 1 ) and then another set by (ikj) = b jt c tik , c m = c (M , (14.2) and consider the transformation scheme We have z J = z 'p( h PJ + c riq *ik = z \ ( b \i + c \ and therefore at the point a tt (ikj) = EUCLIDEAN COORDINATES AT A POINT 19 Now t/ tt = I'^k = eJA;^/,, and the determinant a is equal to the square of thje deter- minant 6, so that the determinant b cannot be zero. It follows that a '^ = e \ ; (H.6) and therefore that is, and therefore (V?)' - - O 4 - 7 ) In this coordinate system the ground form at the point is dx l *+...+dx 2 , (14.8) and the first derivatives of (t ik ... vanish at the point. Of course it is only at the point that these results hold. 15. Two symbols of four indices which are tensor components. Let us now consider the expression We see that since _3 ~ M r MI M k M h Ciczf f (15.1) that is, the expression is a tensor component which should be denoted by T r j c!iiy but as is customary we denote it by (rkhi). (I 5 - 2 ) This is Christotfel's four-index symbol of the tirst kind. We see that if the two first integers are interchanged the sign is reversed, if the last two integers are interchanged the sign is reversed, and if the two extreme integers are inter- changed and also the two middle integers there is no change. The expression a kt z ^ (15.3) is -a vector whose components are tensgr components : it is an associate vector to z. it and may be denoted by z k . 20 TENSOR THEORY We then have z . ri z[ z. llL z( = a li (rkhi). (15.4) Tliis is Christoffel's four-index symbol of the second kind, which should be denoted by J[ rhi > but is denoted by {rthi}. (15.5) Like the four-index symbol of the first kind it is a tensor component. If the last two integers are reversed the sign is changed, so that {rthi} = - {riih}. (15.6) The three-index symbols, it will be remembered, unlike the four-index symbols, are not tensor components. We can express the four-index symbols in terms of the fundamental tensor components ct ik ... and their derivatives. We have (rkhi) = -/A- = Xh-s-ik' + z rh z t ( ijct } > and, as z^ = - (r/7), z^ = ~ (rht), * * V7 Z >'i Z l;~~ l\^ Z >h Z k - Z ii-]th z rh s 1ii> \J tJL/1. ^+*^.^s vJUi >^ __ -^ s >^ ^ *^- ^/ ft ^~ i - P - we therefore have (rkhi) = ^- (rhk) - ^- (rile) + (rit) { kht } - (rht) {ikt}. (15.7) cXi cx h This formula may be written (rkhi) = i(rhk)-h(rik), if we make the convention that the operators are only to act on the last integer, the first two being regarded as fixed,- and the last as a lower integer. We also have {rkhi} = a kt (rthi) = at 1 (I (rht) -h (rit)) = ia kt (rld)-ha ht (rit) = i {rltk}~h {rik} ; TWO SYMBOLS OF FOUR INDICES 21 and therefore {rkhi} = ^- {rhk}- ^- \rik} + {tik} {rht}- [thk] {rit}, *** *** (15.8) since the last integer in {ikt} is to be regarded as an upper integer. It may be noticed that {ikt} (rht) = a*P (ikp) (rid) = (ikp) {rhp}, (15.9) so that in the product {ikt} (rht) the two symbols { } and ( ) may be interchanged. 16. A four-index generator of tensor components from tensor components. If we consider the expression (3-9$ 7*; ?;;;;, ue.i) we see at once that it is a tensor component. To find out what it is we employ Euclidean coordinates at a specified point. At this point we see that it is that is, that is, At the specified point we therefore have w-> = {^>} ^)- {/^%>l (^) ; (iG . 2) and, as this is a tensor identity, it must therefore hold at every point. The proof of this important theorem is a good example of the utility of Euclidean coordinates, at a point. The three- index symbols of Christoffel vanish at any point when referred to the Euclidean coordinates of that point. If they had been tensor components they would therefore have vanished in 22 TENSOK THEORY any system of coordinates. The four-index symbols do not vanish when referred to Euclidean coordinates. The four- ind'ex symbols and the tensor components which are associate to them are the indispensable tools of the calculus when we apply it to differential geometry and to the Modern Einstein Physics. 17. Systems of invariants. We have ( 12) A (a) = a M^u A , and, in accordance with the notion of associate tensors, we may write u k ^ a^ c u. , and therefore A (u) u^u^ (17 . 1) Similarly we have A (u, v) = ti f v t u t v f . (17.2) In accordance with the same notion of associate tensors we might say that u = u ik u . ^ ; (17.3) but this is a rather dangerous use of the notation, as it suggests that the u on the left is the same as the u from which we formed u.^, which is absurd. However, a very moderate degree of caution will enable us to use the Calculus of Tensors without making absurd mistakes on the one hand, or, on the other hand, introducing a number of extra symbols, and thus destroying the simplicity of the calculus, for the sake of avoiding mistakes which no one is likely to make. We have proved, in 6, the formulae -^- a* -f a u { tpq }+a&{tpi} = 0, x p a* = a* {pip}, t and therefore we have -- {pqi}. *^t It follows that ttla'*^ = a*a ik u . ih , (17. 3) / and therefore &2\ u ) a"*^ a*u l . (17 . 4) SYSTEMS OF INVAKIANTS 23 If wo have any invariant of tbe quadratic form a^dx^dx^, say 0, we can obtain other invariants A (0), A 2 (0) by means of the differential parameters; and when we have two invariants, <f> and \Jr, we also have the invariant A (0, \jr). Clearly there cannot be more than n independent invariants. Suppose that we have obtained, in any way, n independent invariants tt 1 ,...u ri . Here the suffixes have no meaning of differentiation or of being tensor components. If \ve take the^e n invariants as the variables, then we have a tk =A(v i u Jl ), (17.5) and we can express the ground form in terms of the in- variants. In this case we can say that the necessary and sufficient conditions that two ground forms may be equivalent that is, transformable the one into the other are that for each form the equations A 0'i Uk) = 0toK---'M'fi) ( 17 - 6 ) may be the same. For special forms of the ground form we may not be able to find the required n invariants to apply this method. Thus if the form is that of Euclidean space there are no invariants which are functions of the variables. IS. An Einstein space, and its vanishing invariants. Let us write A rldh ~ ( rklh) , (18.1) then { rkih} = <^A rpih , (18.2) and therefore (rkil) ^ {rpih}. We form associate tensor components (11) of (rkih)..., and we know that they will be tensor components. Thus we know that a M ( r j^ (18.3) will be a ten or component. We write A space for which all the tensor components A ij ... vanish is .what is called an Einstein space. A space for which 24 TENSOR THEORY where m is independent of the integers r, A, is called an extended Einstein space. We can form invariants from the associate tensor com- ponents. Thus aik A ik (18- 4 ) is an invariant which we may denote by A. Again, d k A ip (18.5) is a tensor component which we may write A k p . We thus have the series of invariants A' P A*, A' P A*AI A' p A*A q r A r t . ... (18.6) All of these invariants vanish for an Einstein space. We can form another series of invariants which do not vanish for an Einstein space. Thus we have }>qra , (18.7) and so on. CHAPTER II* THE GROUND FORM WHEN n = 2 19. Alternative notations. We now consider the ground form a^dx^dx^ for the particular case when n = 2. That is, we are to consider the geometry on a surface which lies in ordinary three-dimensional Euclidean space. The square of an element of length on any surface is given by ds* = a u dx[ + 2a^dx Y dx^ + a^dxl t (19.1) where o n , a 12 , L , 2 are functions of the coordinates x^ x 2 which define the position of a point on the surface. We often avoid the use of the double suffix notation, and take u and v to be the coordinates of a point on the surface, when we write ds 2 = edu? ^ 2fdudv + gdv* ; (19.2) or in yet another form (fe a = A 2 du 2 + 2ABcosotdudv + B 2 dv 2 , (19.3) where a is the angle at any point between the parametric curves, that is, the u curve along which only u varies and the v curve along which only v varies, and Adu and Bdv are the elementary arcs on these curves. There is no difficulty in passing from one notation to the other. The double suffix is the one in which general theorems are best stated : it alone falls in with the use of the tensor calculus which so much lessens the labour of calculation. * [The packets of MS. containing Chapters II and III, as submitted Jo the Delegates of the University Press, were numbered by the author in the reveiso order, and that order would probably have been made suitable, by some reai rangement of matter, had he lived to put the work in readiness for printing. It has seemed best, however, to revert to the order of a list of headings found among the author's papers, an order in which the chapters, as they stand, were almost certainly written.] 2843 E 26 THE GROUND FORM WHEN n = 2 In connexion with the form ch 2 = edu 2 + 2fdudv + gdv 2 we use h to denote the positive square root of egf 2 : that is h = a* = ^5sina, where a = a n a 22 a^. (19.4) The element of area on the surface is hdudv = ABsinadudv = a*dudv. (19.5) 2O. An example of applicable surfaces. If we are given the equation of any surface in the Euclidean space, we can express the Cartesian coordinates of any point on the surface in terms of two parameters and thus obtain e,f, g in terms of these parameters, = &! +2/1+3?, /= i2 + 2/i2/ 2 + 2; i^ 9 = l+yl + *l> (20 . 1) where the suffixes indicate differentiation with regard to the two parameters. Thus, if u is the length of any arc of a plane curve, we may write the equation of the curve y = <f> (u), and the surface of revolution obtained by rotating the curve about the axis of x will have the ground form where v is the angle turned through. Can we infer that, if a surface has this ground form, it is a surface of revolution ? We shall see that we cannot make this inference. Thus consider the catenoid, that is, the surface obtained by the revolution of the catenary about its directrix. The ground form is fa* _. c j u z + ^2 + c ^ j v ^ Take the right helicoid, given by the equation z = x this is clearly a ruled surface, and we can express the coordinates of any point on it by i x = u cos v, y = u sin v, z = cv* AN EXAMPLE OF APPLICABLE SUKFACES 27 Its ground form is then and it is not a surface of revolution. It is, however, applicable on the catenoid ; two surfaces which have the same ground form being said to be^applicable, the one on the other. 21. Spherical and pseudospherical surfaces. The tractrix revolution surface. There are two distinct classes of theorems about surfaces : there are the theorems which are concerned with the surface regarded as a locus in space ; and there are the theorems about the surface regarded as a two-way space, and not as regards its position in a higher space. It is the latter type of theorems about which the ground form gives us all the information we require. Thus all the formulae of spherical trigonometry can be, as we shall see [in the next chapter], deduced from the ground form <; 6 2 _ rfu 2 + S ; n 2 u ^ (21.1) where u is the colatitude and v the longitude. We shall prove the fundamental formula cos c = cos a cos b + sin a sin b cos (7, (21 . 2) and the formula for the area 4 + .B-K7-7T, (21.3) and from these all the other formulae may be deduced. So from the ground form cL 2 = du 2 + sinh 2 udv* (21.4) we can obtain the formulae of pseudospherical trigonometry the trigonometry on a sphere of imaginary radius. The fundamental formula is here cosh c = cosh a cosh b sinh a sinh b cosh C, (21.5) and the area of a triangle is n-A-B-C. (21.6) If in (21 . 4) we make the substitution (c being a constant) u = u f c, v = 2e~ e v' 9 this ground form becomes 28 THE GROUND FORM WHEN n = 2 and if we take c to be a large constant it approximates to the ground form ( / 8 2 _ du z + e -^dv 2 9 (21.7) and to this form pseudospherical trigonometry will also apply. The formulae of spherical trigonometry or of pseudo- spherical trigonometry will apply to any surfaces which have the same ground form as the sphere or the pseudosphere. A real surface may have as its ground form d** = du* + e-* u dv\ Thus if we take a tractrix, the involute, that is, of a catenary which passes through its vertex, the equation of the catenary is y _ cos h x , (21. 8) taking the directrix of the catenary as the axis of x ; and if we take u as the arc of the tractrix, measured from its cusp, the vertex of the catenary, the equation of the tractrix is y = e- u . (21.9) If we now revolve the tractrix about the axis of x we get a surface of revolution with the ground form d6* = du* + e-* u dv*. (21 .7) The figure of the tractrix is something like Fig. 1 ; and its surface of revolution like Fig. 2. 22. Ruled and developable surfaces. The latter ap- plicable on a plane. Let us now consider the most general ruled surface, formed by taking any curve in space as base, or as we shall say as directrix, and drawing, through each point of the directrix, a straight line in any direction determined by the position of the point on the directrix. If x, y t z are the coordinates of any point on the directrix, and I, m, n the direction cosines of the line, then [these coordinates of the point and these direction cosines of the line will be functions of a parameter v. We take u to bo the distance of any point on the line from the point where the line intersects the directrix. Then the current coordinates of any point on the line may be written a = x 4- ul t y' = y + um, z' = z + un ; Fio. 1. Fio. 2. 30 THE GROUND FORM WHEN H = 2 and for the ruled surface we have the ground form d* 2 = du 2 + 2fdu dv + gdv 2 , where (22.1) ' 2 + m^ (22.2) That is, / is a function of v only and # is of the form where a, /3, y are functions of v only. We have ds 2 = (du +/cZy) 2 + (g -/ 2 ) cfe 2 ; (22.3) the coordinates of any line of the ruled surface are functions of v only, and therefore the shortest distance between the point u y v and a neighbouring point on the line whose coordinates are functions of v + dv is (gf 2 ) dv 2 . The value of u for which this shortest, distance will be least is then given by -^ = ; that is, the equation of the oil/ line of strict! on is ^ n V - = 0. (22.4) dl6 V ' If we take, as we may, the directrix to be a curve crossing the generators at right angles, and dv to be the angle between two neighbouring generators, we have ds 2 = du 2 + ((u - a) 2 + b 2 ) dv\ where a and b are functions of v. The line of striction is now u = a, and the shortest distance between two neighbour- ing generators is bdv. For a developable surface therefore we have d* 2 = du 2 + (u- a) 2 dv 2 . (22.5) If we take u' = u sin v la cos vdv f v' = u cos v | a sin vdv, (22 . 6) we see that referred to the new coordinate system (22,7) so that the above transformation formulae establish a corre- RULED AND DEVELOPABLE SURFACES 31 spondence, between the points on any developable and points on a Euclidean plane, such that the distance between neigh- bouring points on the developable and the distance between the corresponding neighbouring points on the plane are the same. The developable is therefore said to be applicable on the plane. 23. Elliptic coordinates. Consider now the system of confocal quadrics 992 * 2 , y _ ,_ _?* a 2 -f u b 2 + u c 2 + '16 We know that the relations 2 _ (a 2 -f u) (a 2 + v) (a 2 + w) 2 __ (b 2 + u) (b 2 -f v) (b 2 4- w) ( ' ' give the coordinates of any point in space in terms of the focal coordinates u, v, w ; and that the perpendiculars from the centre on the threo confocals through any point are given by 2 _ - 2 = ~~ ' ~~ (v u)(vw) 2 4- w) (b 2 4- w) (c* + w) 2 __ ~~ w uw v From the formula p* = (a 2 + u) cos 2 a + (6 2 + u) cos 2 + (c 2 + u) cos 2 y, (23. 2) where x cos a + y cos /? + z cos y = p is the tangent plane to the surface u = constant, we see that = du, 2qdq = dv, 2rdr = dw, and therefore 4 cfe a = + + - (23 . 3) p^ g 2 r^ If we now take w = and write * U\ a 2 + v=F 8 , K{^^~b\ K\ = a 2 ~c 2 , 32 THE GROUND FORM WHEN H = 2 so that the new coordinates are the semi-major axes of the two confocals through any point on the ellipsoid we Jmve * y *-K*J K*)(U'-K*) (V*-Kl)(V*-K*) (23.4 It follows, as a particular case, that the ground form for a plane may be taken to be ,.,-, .^L. ,. We thus have as ground forms of a plane d* 2 = dx 2 + dy 2 , / 2 2\ = (u 2 - v 2 ) ( -~ 5- -f - ' and we could find an infinite number of other forms for the plane, or for any other surface. We are thus led to inquire as to the tests by which we can decide whether two given ground forms are equivalent ; that is whether by a change of the variables the one form can be transformed into the other. 24. The invariant K. A0 and A 2 when K is constant. Consider the form a ., ^ ,7^ _ t o a ik ax i (lx k) ^ *> ^> and let us use the methods of the tensor calculus. In terms of the four-index symbols of Christoffel we have one and only one invariant (1212)-ra,* (24.1) where a = a n ^ 22 ~ (^] 2 ) 2 ' * [The invariants A of (18. 4) reduce to one. Also, as the equalities (1212) = -(2112) = -(1221) (2121) hold, and the other symbols (1112), &c., vanish, the sum equal to (1212)' in (15 . 1), with the notation of (15 . 2), is For the explicit expressif n of K see Chap. Ill, 43.] THE INVAIUANT K 33 We denote this invariant by K. ^ Let us first take the case when K is a constant and con- sider the differential equations (24.2) We shall prove that they form a complete system : that is, a system such that no equation of the first order can be deduced from them by differentiation only. We have . m + A' u & = 0, <f> . m + Ka^ = 0, and therefore [ 16] that is, ~{H 12] (f> f + K (a ll 2 - la 1 ) = 0. (24.3) We have then to show that this is a mere identity. Now {It 12] = a f l'(l2) 12) = u t2 and therefore {1*12} fc = so that the equation of the first order turns out to be a mere identity. Similarly we see that the other equation of the first order is a mere identity. If and -v/r are any two integrals of the complete system we have d i ( p - a * k (0 . ip + k + 0^ . Jq} ) + K = 0. (24.4) We therefore have A (0) + K 2 = constant. (24.5) We also have at once from (12.5) and the equations (24 , 2) A 2 (0)+2jff0 = 0. (24/6) 34 THE GROUND FOKM WHEN U = 2 25. Determination of a \jr such that A (0, \/r) = 0. We shall now prove that if we are given any function u, such that A(u) and A 2 (u) are both functions of u t then, in all cases (not merely when K is a constant), we can obtain by quadrature a function v such that A (w, v) = 0. Let J*W*u H = e J A W , (25.1) , , A 2 (u) then *=-*. I The condition that where u 1 == a 11 u 1 may be a perfect differential is that is, fi a* A 2 (u) -f a* (//j u 1 -f /^ 2 u 2 ) = ; and this condition is fulfilled. We can therefore by quadrature find a function v such that v 1 = / ua^u 2 , Vgsr-^aht 1 , (25.2) and therefore ^u 1 -f v 2 u 2 = 0, that is, A(u, v) = 0. (25.3) 28. Beduction of a ground form when K is constant. Returning now to the case when K is a constant, we have seen that, if is an integral of the complete system, A 2 (0)4 2#0 = 0, A ($) + /f< 2 = constant, and we can therefore by quadrature obtain \fr t where A(0,V) = 0. First let us take the case when K is zero. Without loss of generality we may suppose that A(0)=l, A(0,^)=0, (26.1) and we may take as new variables x l = 0, aj a = ^r, and the ground form becomes cZs* = dx\ 4-tf 22 efce*. REDUCTION OF A GROUND FORM WHEN K IS CONSTANT 35 Since x l is an integral of the complete system, we have {111} = 0, {121} = 0, {221} = 0, . and therefore (111) = 0, (121) = 0, (221) = 0. From the fact that a 12 is zero, we have (122) + (221) = o, and therefore (212) = ; so that a 22 is a function of OJ 2 only. We can therefore take the ground form to be cZs 2 = dx\+dx*. (26.2) We next take the case when K is a positive constant, say R-*. We then have A (0) + jR~ 2 2 = constant, and, without loss of generality, we may suppose A(0) = ,R- 2 (l-0 2 ), (26. 3) and, by quadrature, we can find ^ so that A (0, f ) = 0. (26 . 4) Take as new variables x^ R cos' 1 0, # 2 - i//-, (26 . 5 ) and the ground form becomes cZ& 2 = dx\ +a t22 dx%. We have, since a 12 is zero, (122) + (221) = 0, /> and, since cos-~ satisfies = 0, we have (221) + ^p c ot( J) = ; and therefore (2 1 2) = cot that is, ~a 22 = 2a 22 36 THE GROUND FORM WHEN U = 2 so that sin 2 ( fl) C/ 22 is a function of rr 2 only. We may therefore take the ground form as (/6- 2 = dxi + sin 2 -,~ dx'] , or if we take ^ = Rx\, x\, = ^/ 2 5 we may take the ground form as ds 2 = 2 (dU-J + sin 2 ^rficS). (26 . 6) When K is a negative constant JK~ 2 , we see that the ground form is ^ _ -2l 2 (dxl + sin 2 0^/0;:;) ; (26 . 7) or if we take x\ LX V x' 2 a\ 2 , the ground form becomes da* = Ji 2 (//^'J +sinh 2 a; 1 ^.]). , (26 . 8) We have seen in 21 how the ground form dtp = R* (dxi \-e~"**dx\) (26 . 9) may be deduced from this. 27. The case of A (K) = 0. We have now seen the forms to which the ground forms are reducible when the invariant K is a constant; and we see that the necessary and sufficient condition that two ground forms may be equivalent, when for one of them K is a constant, is that for the other K may be the same constant. We must now consider how we are to proceed when K is not a constant. If A (K) is zero, we choose as our variables x l = K, x> 2 = v, where v is any other function of the coordinates of the assigned ground form. Since A(a; 1 ) is zero, a 11 is zero and the ground form may be written dip = edu <2 + 2(f) 2 dudv y (27.1) where e and </>., ( = -- ] are some functions of u and v. THE CASE OF A (K) = 37 The equation which determines the invariant K which we have taken to be u is therefore or (2 02 __ 2u(/>) = 0. (27.2) We may therefore take e 2o where a and /? are functions of u only. The ground form now becomes, if we take x l = K, x 2 == 0, efe 2 - (ajjoj^ + otx 2 + /3) dx\ + 2 cfoy7.r 2 , ( 27 3 ) where a and /3 are functions of a; x only. We can then decide at once whether two ground forms for each of which A (K) is zero are equivalent. 28. The case when A 2 7i and AK are functions of K. We may now dismiss this special case when A (K) is zero: it is not of much interest, as it cannot arise in the case of a real surface. We now consider the case when K is not a constant and A (A") is not zero, but A 2 (K) and A (K) are both functions of /{". This arises when the surface is applicable on a surface of revolution. Let us take u = K t (28 . 1) and let v be the function which we have seen can be obtained by quadrature to satisfy the equation A(u,v) = 0, (28.2) when A 2 (u) and A (u) are both functions of u t though the reasoning would have held equally had A 2 (u) ~- A (u) been only assumed to bo a function of u. We saw that if f A * (n \/ ?t 38 THE GROUND FORM WHEN H = 2 If then we take 6 = \idu^ we' have () l = pu^ 2 = /zi^. and therefore ^ = a! (a 12 ^ + a 22 2 ) = a*0 2 . Similarly wo have v 2 = alQ 1 ; and therefore t = a^v 2 , 2 = alv 1 . It follows that tf^ 1 = v 2 v 2 , 2 2 = 14 v 1 , and therefore A (fl) = A (v), A (0, v) = 0. (28 . 3) We also have (c fl) 2 = (yucJu) 2 , A ((9) = (/z) 2 A (tt) s and therefore ^ = ^- (28.4) A((?) A(u) 7 If we now take as the new variables 6 and v, the ground form becomes ^2 dv z A ((9) ' We therefore see that the ground form may be written fA ? (* J A W rf 2 = (A (A^)- 1 ((dKf + c J A W Jv 2 ), (28.5) where v may be expressed by quadrature in terms of K and integrals of functions of it. We thus see that given two ground forms, for each of which A (K) is a function of K and also A 2 (K) is a function of K, the two forms are equivalent if, and only if, the functional forms are the same. 29. Conditions for equivalence in the general case. Finally we have the general and the simplest case when K is not a constant, and A (K) and A 2 (K) are not both functions of K. In this case we have two invariants, say u and v. We CONDITIONS FOR EQUIVALENCE IN THE GENERAL CASE[ 39 take these invariants as the coordinates, when the ground form becomes , 2 __ A (v) (fa, 2 - 2 A (u, v) dudv 4- A (u) dv* ~ * ' ( } The necessary and sufficient conditions, that two such ground forms may be equivalent, are that, for each of the forms, A (w), A(u, 9 v), A (v) (29.2) may be respectively the same functions of u and v. We now know in all cases the tests which will determine whether two assigned ground forms are, or are not, equivalent. 30. The functions called rotation functions. When the measure of curvature * is constant we saw [ 24] that the re- duction of the ground form to its canonical form depends on finding an integral of the complete system of differential equations 0. n + /fa u = 0, <f>. n + Ka n (f) = 0, (/> . 22 + Ka 22 (f> = 0. (30.1) We shall now show how this integral may be found by aid of Riccati's equation. Take any four functions, which we denote by q l9 y 2 , ? 1 19 r 2 and which will satisfy the three algebraic equations Ka u = ql +rj, Ka u = qfa + r^, Ka^ = q$+r*. (30 . 2) The functions thus chosen are not tensor components, but we shall operate on them in accordance with our notation with 1 and 2. These two operators annihilate Ka n , Ka l2 > /ia 22 , and there- fore we have + r l r 1>a = 0, We define two other functions p l and p 2 by * [This name for the invariant K will bo shown later to have geometrical fitness. See 37.] 40 THE GROUND FO11M WHEN H = 2 It at once follows by simple algebra that ?V 2 + ?A,<7 2 ^= 0, </ 2 . 2 -j>V'2 = We then havo i^ + ^iV/i +7V/1-2 = 0, and therefore (21 12)?^ + q l (p } . 2 ?Vi) = 0, that is, {1^12}^= <h(Pi-2-P*-i)> (30.4) or a tk (1 i 12) r/ = ?1 (j> 1 .,-^ 2 . 1 ). (30 . 5) Now a tk (1 fc 12) r f = a' 2 (12 12) -^ = and therefore Similarly we luive (30 - 8) Aay six functions ^ , ^; 2 , ^ x , <7 2 , r ls r, 2 which satisfy these three equations are called rotation functions. They havo important geometrical properties and are much used by Darboux, but here we simply regard them as algebraically defined functions. 31. Integration of the complete system of 24. Now consider the equations A U = 0, v -- mr 2 + 7ij a = ; ' c' X' / 7 0, ^- - ^ a + Zr 2 = ; INTEGRATION OF THE COMPLETE SYSTEM OF 24 41 These equations are consistent because the functions Pi>Pv Vi,<l2> r i> r 2 are rotation functions ; and we see that I 2 -f- m 2 -f n 2 is a constant. Let q. = * + '_. , (31.2 V^ 2 + m a + 7t 2 -/J, where stands for \/ 1 : wo have We can therefore find or by the solution of Riccati's equation. To determine Z, in, n we have l + im , l im cr = "" and f 2 + m 2 + 7i 2 = constant : and we thus see how, when we are given the rotation functions, we can determine I, m, n. We can now at once verify that l. u + Ka n l = 0, I. l2 + Ka 12 l = 0, l. 2Z + Ka zz l = 0, (31.4) and that A (1) = K (m* + n 2 ) = K (constant - 1 2 ). (32.5) We have thus shown how a common integral of the com- plete system can be obtained by aid of Riccati's equation. The I, m, n which we thus obtain will be the direction cosines of the normal to the surface, but we do not make any use of our knowledge of a third dimension in obtaining I, m, n. CHAPTER III* GEODESICS IN TWO-WAY SPACE 32. Differential equation of a geodesic. We have now considered the ground form of a surface, and wo know the method by which we are to determine when two given ground forms are equivalent; that is when they are transformable the one into the other by a change of the variables. We now wish to consider the geometry on the surface regarded as a two-way space ; and we are thus led to the theory of geodesies. We have ds 2 = UikdXidXfa (32.1) and therefore dSs __ dx A dSx k dx k dSx { cfo. dx k 2 7/iT "" ''* ds ^T + ik di ds ' + ds "57 ik ' d <fai <lx k d,. fe T ~T~ ; ' ~ : ^ - OX*. ds ds dx t For a path of critical length we therefore have d / </./-,. \ d/ dx,.\ <*(/*,. dxjdx,. ~T ( a it ~j ) + -/-( a th i ) = -^ ,r-' (32 . 2) ds\ H dd/ ds\ tlc ds / dz t ds ds v ' Now ^ = (/tt) + (*); o^ and we notice that, though (itk) ^ (kti), yet ^ ^ fc __ a . dx 4 dx L (Uk} 'ds lfc- (Ul >~ds'~ds (32 ' 3 ^ * [See foot-noto on p, 25.] DIFFERENTIAL EQUATION OF A GEODESIC 43 The path of critical length therefore satisfies the equation (/tfA. '(32:4) ' 33. Another form of the equation. The expressions dx. , flx>, -- 1 and ~ a 6- ds are called the direction cosines of the element. They determine the position of the clement at the point x ly # 2> but they are not, in general, the cosines of the angles the element makes with the parametric lines. We denote them by l , 2 in tensor notation. For a geodesic wo therefore have We can put this equation in another form. We have and therefore " it Now gp (tpi) = ^^ (tki) = g k (kti) and pg* (Ltk) = ggt (kti). We then have for a geodesic and therefore, multiplying by a!*! and summing, For a geodesic we thus have either of the two equivalent equations ,/ '^ = 0. (33.4)' 44 GEODESICS IN TWO-WAY SPACE 84. Condition that orthogonal trajectories be geodesies. If two elements at x 19 x 2 are perpendicular to one another we* have a ll dx l Sx l -f a ]2 (dx v 8x^ + dx^Sx^ + a 22 dx 2 8x 2 = 0. (34 . 1) The elements perpendicular to the curve = constant will then satisfy the equation 2 (a n dx t + a l2 dx t2 ) (f> l (a^rf^ +tt 22 tZoJ 2 ), (34.2) or "n^ + f^ 2 = M> ^ 1 + "22^ 2 = ^. ( 34 - 3 ) where /z is some multiplier. We have fi = ,1 (^0,-f a 12 2 ), ^ - ^ (^ l 1 + f( M a ), (34 .4) and, as {&**= ] we see that // f^ 1 X -f ^ 0J = 1 ; and therefore /^ 2 A (0) = 1. , (34.5) We thus have L-, WW '+ M ^ = _ , (34. (0) -/A (0) 7) ' ; Now '~(a u g)-(i ORTHOGONAL TRAJECTORIES 45 And, as A (<j>) = aPV<l> p <j> q , we see that (A (<j>)) t = We therefore have . 2A(0)~~ 2(A(0)) 2 > Suppose now that is a function of the parameters such tbat A (0) = F($). (34 . 9) We see at once that the right-hand member of the above equation vanishes; and therefore the orthogonal trajectories of the curves <f> = constant are geodesies, if A (0) is a function of 0. Conversely we sec that, the orthogonal trajectories of any system of geodesies being (f) = constant, A (0) must be a function of 0. 35. Geodesic curvature. Jf l , 2 are the direction cosines of an element of the curve constant, we have ft? - . <'ikt't k = i, and therefore & = a*(A0)* 2 , 2 = -a*(A0)*'. (35.1) Differentiating P<f> = with respect to the arc, we have that is, < / > (T;' > + {ikp} l j + Q'pq^^ = - We therefore have, summing along the curve, 1 + { ^ 1 } tf^ds) ^ (d^ + f. + (35.2) The first integral if summed along a small length of the curve only differs by a small quantity of the second order 46 GEODESICS IN TWO-WAY SPACE from the same integral if summed along the curve formed by the geodesic tangents at its extremities. Now, summed along a geodesic, wo know that the first integral vanishes, but summed along the curve formed by two geodesies the integral is IV-IV, (35.3) where l , 2 and rj 1 , ?? 2 are tho direction cosines of the two geodesies at their common point. The angle a between the two elements whose direction cosines are l , 2 and rj l , r? is given by a*(V-V) = sina. (35.4) We therefore have tho formula a ~^ = (^^ )2 ~ 2 ^ (35.5) where dO is tho small angle at which the geodesic tangents at the extremities of ds intersect. The formula for the geodesic curvature of the curve </> constant is therefore ^M0Mi(W a -20M20i& + 0. M (0,)>-Wr f . (35-6) 36. We can express the above formula in a better form : to prove this we employ the coordinates which are Euclidean at a specified point. We have at tho specified point and therefore Now A 2 (0) = u + 22 , and therefore at the specified point ,A0, (36.2) and at the specified point a is unity. GEODESIC CURVATURE 47 We thus have at the specified point of the curve, and there- fore at every point of the curve, > (36 - 3) The method of thus employing Euclidean coordinates is very helpful in proving formulae in the tensor calculus. The direct proof of the equality (36.4) would be much longer. 37. Polar geodesic coordinates. The measure of curva- ture K. The geodesic curvature of a curve is given by the formula + {ik2} *' (37.1) where g l t 2 are the direction cosines of an element of the curve. If we take, and we shall see that we can take, the ground form of the surface to be du* + l?dv\ (37.2) where u is the geodesic distance of any point on the surface from a fixed point on the surface, and the curves v = constant are geodesies passing through the fixed points, dv being the angle at the point between two neighbouring geodesies, jB being a function of u and v which on expansion in the neigh- bourhood of the fixed point is of the form tt-f ..., where the terms denoted by +... are of degree above the first, we employ what we may call polar geodesic coordinates with respect to the fixed point. Let us now employ polar geodesic coordinates to interpret the formula for geodesic curvature. We have {111}=0, {112} = 0, {121} = 0, {122}=^, {221} = -/?!, and {22f2}=jp. (37.3) 48 GEODESICS IN TWO-WAY SPACE If d is the angle at which the curve crosses the geodesies through the fixed point, and [see 43] the measure of curvature K of the surface is given by KB + B n = 0. (37.4) We have and therefore = -, h -W- sin 0. (37.5) A/ ds B ^ ' Now consider the expression (37.6) where dS is an element of area of the surface, and take the summation over the small strip bounded by two neighbouring geodesies through the origin of the polar geodesic coordinates and an element of the curve. The expression is KBdudv, and this is equal to B u dudv r 7? = dv- U^sintfds. (37.7) rds It follows that Jpg taken over the boundary of any closed curve surrounding the point is equal to <. pp P H (37.8) MEASURE OF CURVATURE 49 where the integral is to be taken over the area of the curve. We thus have a geometrical interpretation of the measure of curvature: it is the excess of 2?r over the angle turned through by the geodesic tangent, as we describe a small closed curve, divided by the area of the curve. It will be noticed that in this definition we do not make use of any knowledge of a space other than the two-way space of the surface itself. This is what the curvature of a two-way space must mean to a mathematician to whom the knowledge of a three-way space can only be apprehended in the same vague wa} r as we speak of a four-dimensional space. 38. Recapitulation. Parallel curves. It may be con- venient to bring together the various formulae which so far we have proved in connexion with direction cosines and geodesies before we proceed further. ds * ds ' where the direction cosines are those of an element of the curve = constant ; (34 '.7)' where the direction cosines are those of an element perpen- dicular to the curve ; for a geodesic we have d , ( I' / = 0- ( 33 *)' The orthogonal trajectories of the curves (f> = constant, where A (0) is a function of 0, are geodesies, and the ortho- gonal trajectories of any system of geodesies are curves (p = constant, where A (0) is a function of (f>. (34 . 9)' Leaving aside the case when A (0) ^is zero, we can choose the function so that A (0) = 1. 2843 II 50 GEODESICS IN TWO-WAY SPACE If we know any integral of this partial differential equation involving an arbitrary constant <f>(x l9 x 2 , Oi) = 0, then the system of curves /J oft will be geodesies : for A ( 0, ^~ J = 0, and, as the condition that two families of curves <p = constant, ty = constant, may cut orthogonally is A (0, ty) = 0, we conclude that the CUrV6S ^ = (38.1) da will be geodesies, since they cut the curves = constant orthogonally. If we choose the arbitrary constant /? so that the geodesies given by ^0 ^ d a"' 3 may all pass through a fixed point, and if we take the equation of the geodesies to be v constant and take v as one of our parametric coordinates and to be the other parametric coordinate u, we have A(u) = 1, A(tt, v) = 0. The ground form of the surface then takes the form du* + B 2 dv*, (37.2) and in the neighbourhood of the fixed point, through which the geodesies pass, we may clearly take from elementary geometry that jB = 7t + . . . . We thus have what we called the polar geodesic coordinates. We have d<j> = <f> l dx l -\-(f> 2 dx 2) and therefore, l , 2 being the direction cosines of an element perpendicular to the curve = constant, the length dn of the normal element is given by d<f> = ~ or * = ,/A70). (38/2) RECAPITULATION. PARALLEL CURVES 51 The curves which satisfy the equation A ((/>) = 1 (38 . 3) are called parallel curves. We see thus that two parallel curves cut off equal intercepts on the geodesies which cut them orthogonally. If particles, constrained to lie on a smooth surface and acted on by no forces but the normal reaction of the surface, are projected at the same instant, with the same velocities normal to any curve, they will at any other instant lie on a parallel curve. From the theory of partial differential equations we know that any curve on the surface will have a series of curves parallel to it, though the finding of them involves the solution of the equation A (0) = 1. The explicit forms of the differential equations of a geodesic are a l + {lll}xl+2 {121} 1 * 2 +{221}aj;j = 0, a : 2 -f{112}a'i+2 {122} 0^2+ {222} a* = 0, (38.4) where the dot denotes differentiation with respect to the arc. If we write the variables as x and y and let dy (Py = = we have y = xp t y and the equation of the geodesic becomes </- {221 }>'' + ({222} -2 { 121 })^ 2 + (2 {122} _ {]!!})>+ {112} = 0. (38.5) 39. Notes regarding geodesic curvature. Now consider- ing geodesic curvature, in the figure on p. 52 P and Q are two neighbouring points on any curve, PT and TQ are the geodesic tangents at P and Q, and QM is an element of arc perpendicular to the geodesic tangent JPTM. By definition the geodesic curvature of the given curve at 52 GEODESICS IN TWO-WAY SPACE P is the limiting ratio of the angle QTM to the arc PQ as Q approaches P. We therefore have = 2 /|?f> ( 39 -o Pg J- W and thus have the analogue of Newton's measure of curvature of a plane curve for a curve on the surface. It is the geodesic curvature only that has a meaning when we conline our attention to the two-way space on a surface. We have the formula V (36 . 3) Pg VA(0) ' ^" " ' and we may apply it to find the geodesic curvature of the curve all the points of which are at a constant geodesic distance from the origin, in the polar geodesic coordinate system. We have ds 2 du? + K*dv\ = u, and therefore = r- log B. Pg ^ The curvature will be constant if, and only if, B=f(u)F(v), that is, if the surface is applicable on one of revolution. The curvature will then only bo - , as it would be in i * u a plane, it -, 9 72,972 1 ' ds 2 = du* + u*dv*, that is, if the surface is applicable on a plane. If we take, the case where K is positive unity and da 2 = du 2 -f sin 2 u dv 2 , yve see that the geodesic curvature of a small circle is cot u. If we take the form NOTES REGARDING GEODESIC CURVATURE 53 which is applicable to the tractrix or any surface applicable on it, wo see that the geodesic curvature of the curves u == constant is minus unity. 40. The formula for the geodesic curvature may be written (40 n Let fji be an interating factor of where 1 = tt n 0,+a 12 2 , 2 = so that and therefore , - (/za0 ] ) -f r-~7 (/zc^0 2 ) = 0. O&'i O &'.,, Now a*A 2 (0) - ^0', A (0, /i) = <pn t , and therefore /z A 2 (0) + A (0, /z) = 0, that is, A. 2 (0) + A (0, log /z) = 0. The formula for curvature may therefore be written (4() This is an equation to give the integrating factor. When the integrating factor is known we can find the function -v/r by quadrature ; and, as A (0,^)= 0^=0, (40.3) we have then the equation of the orthogonal trajectories of the curves = constant. In particular when the curves = constant are geodesies, we may take ^ (A0)~i, (40 . 4) and we thus see that the orthogonal trajectories of any system of geodesies may be found by quadrature. In eneral we have and thus the formula for the geodesic curvature, of the curves = constant may be written (40.5) Pg where the curves ^ = constant are the orthogonal trajectories. 54 GEODESICS IN TWO-WAY SPACE 41. Integration of geodesic equations when K is con- stant. We have obtained the differential equation of a geodesic on any surface, but, in general, we cannot solve the equation we have arrived at. Sometimes we can. Thus when the measure of curvature is positive unity we may take the ground form to be dt p _ du * + sin 2 ^ dv ^ ( 4 1 u } j We then have as the equation of the geodesic d6 that is, cos -,- + cot u sin 6 = 0, au or sin 6 sin u = constant. XT A ^ JMow sin = smu-j-) as (ID and therefore sin z u ,~ = sin a, (41.2) where a is some constant. We could have obtained this equation directly, as we easily see, by the rules of the Calculuy of Variations. We deduce that and therefore cos u = cos a cos s, and we thus obtain the equations sin 8 sin a cos s . tans . . sm v = - , cos v = . ------- , tan v = -.- (4 1 . 3) sin u sm u sin a We now see that cos 14 cos u 2 -f sin Uj sin u 2 cos (v l v 2 ) = cos 2 a cos Sj cos 8 2 -f sin 2 a cos s x cos s 2 -f sin ! sin s 2 , = COS (^ Sg). This is just the well-known formula of spherical trigonometry cos c = cos a cos 6 + sin a sin 6 cos (7. (41 . 4) Similarly we could obtain the formula cosh fo f 2 ) = cosh *j cosh u 2 sinh ^ t sinh u 2 cos (i^ v 2 )/ (41.5) INTEGRATION OF GEODESIC EQUATIONS 55 which would be applicable to a surface of constant negative curvature. The formula when applied to a geodesic triangle on a surface of curvature positive unity, gives us the well-known formula for the area of a spherical triangle A+B + CTT (41.6) and more generally for any surface of constant curvature TT). (41 . 7) 42. Focal coordinates. If we take, as the coordinates of a point on a surface, tho geodesic distances of the point from two fixed points on the surface, the ground form will take the form ( s i n a )-a (^2 + dv 2 __ 2 cos a dudv), where a is tho angle between the two geodesic distances. We easily see this geometrically, using the property that the locus of a point at a constant geodesic distance from a fixed point is a curve cutting the geodesic radii vectores orthogonally. Analytically we prove the formula from the fact that A (u) and A (v) are both unity, and applying this to the general ground form A 2 ( In 2 + B 2 dv 2 when we have A 2 = B 2 = cosec 2 a. If we take 2x we have db 2 = sec 2 -- dx 2 -f cosec 2 - dy 2 . (42.1) 2 6 This system of coordinates may be called focal coordinates : the curves x = constant will represent confocal ellipses ; that is, curves the sum of whose geodesic distances from two fixed points, which we call the foci, is constant. Similarly the curves y = constant will represent confocal hyperbolas, and we see that the ellipses and hyperbolas intersect orthogonally. .43. Explicit expressions for symbols {ikj} and for K. It will be convenient here to give in explicit form Christoffers 56 GEODESICS IN TWO-WAY SPACE three-index symbols of the second kind,* as we so often need them, and expressions for the measure of curvature. "We take the ground form da 2 = cdu 2 -f 2/dudv + gdv*, (43.1) and we then have a == It 2 egf 2 , (43 . 2) * 2 -2/ 1 ), 2/^(112} = =tf(2/ 2 -fl 1 )-/ S r 2 , 2 /, 2 {222 } = */ 2 (43.3)... (43.8) 4 A 4 # = e (r/ a ( 2 - 2/J -f gl) + flf (^ (^ - 2/ 2 ) + e ) If we take as the ground form the last formula becomes M n . ^ /R ul () cosa\ ^ /An B, cosa\ A B sin a A + a, 2 -f ^-- ( . .- -- ) + ^ ( Si~ ' - 1 = 0, 1Z CU\ ^i Bin OK / CV \ ,0.3111 OK / (43.10) which is Darboux's form. * [Those of tho first kind arc at once (HD-i^, (112) -^-ic (121) (211) = J Ca , (122) (212) (221) -/.H^, (222) = J^ 2 . Wo also have -h flf ( - Cr ^-- f <> 2 EXPLICIT EXPKESSIONS FOB SYMBOLS 57 In particular if the parametric coordinates are geodesies we have {221} = 0, {112} = 0, (43. ;i) and therefore rt A 9 ~B, cos a 2 A sin a ' x B sin a ' and the formula for the curvature takes the simple form ABsin aK = 12 . (43 . 12) From this formula wo could easily deduce again the formula r*uffA'ci=2,r. J p^ JJ When we take the ground form to be 2 we have When we take the ground form to be we have {111}=, {112} -> {121} =-' 2 -, [122}=* 1 j 2e ( 2e ^ 2e *- * 2e (221 } = ~, {222} = - 2 -, 2e 2e (43. 14) ...(43.19) ; 0, (43.20) (43.21) + (!#)> (43.22) Finally, when we take the ground form to bo 2fdudv, 58 GEODESICS IN TWO-WAY SPACE ^ 44. Liouville's special form. When the ground form of a surface takes the special fdfm Liouville's form 17 and F denoting functions of u and v respectively, we can find a first integral of the equation of geodesic lines. For the form e ci l = cos 0, ei 2 = wind, and the equation of a geodesic becomes ^0 . A 2 e* + e : sin - 0o cos Q ~ 0, that i, - (cl sin ^) + v (& cos 5) = 0* (44.2) c l(j v V We therefore have e^ sin 6 = 0o, c^ cos = 0j , and e = 0i -f </>;;, that is, A(0) = 1. (44. 3) In the particular case of Liouvillc's surface <t>i-U=V-M, and \ve obtain a complete integral of this partial differential equation by equating the above expressions to a constant. We thus have ^ = ^(j~+a t <P, = SV~-a, giving the first integral e^cos^ = VU+a y 6*siii tf = -/F a, (44.4) <^ 2 '^ 8 /^, r, or y 7 - = TT -- (44.5) 45. Null lines. Complex functions of position. We shall now consider a further application of Beltrami's differ- ential parameters to the geometry of surfaces. NULL LINES. COMPLEX FUNCTIONS OF POSITION 59 The null lines of a surface arc the lines which satisfy the equation a ik dx^lx k = 0. (45 . 1) These lines play the part in the geometry of a surface which the circular lines play in plane Euclidean geometry. If the equation of the null lines is <f> = constant, then A (0) = 0. (45 . 2) To obtain the null lines wo must therefore be able to solve this equation. The equation will have two independent integrals. If we take these integrals as the parameters we employ what we call null coordinates. The ground form takes the form 2fdudv, (45.3) j i, x /*\ 2 ^0 ^0 and, as we have seen, A (<p) = - --- --- - If then A (0) = 0, must be a function of u only, or a function of v only. A function satisfying the equation may bo called a complex function of position. There are therefore only two types of complex functions of position, viz. the two functions whose differentials are multiples of the factors of d&. The first we shall take as that which corresponds to the factor (a n dx l + (a 12 + i Va) dx^ -f a n , (45 . 4) and the second that which corresponds to (a ll rfo; 1 -f ( 12 ' Va) dx^ -f- a n i. (45 . 5) We need only consider those which correspond to the first factor, and, if we do this, we can say that every function of position is a function of every other such complex function. Thus in the case of the plane, where we have rfs 2 = dx z + dy* = dr* + r*dO*, x + LIJ is a complex function of position since its differential is a multiple (unity) of dx+idy of the first factor of dx^ + dif, and log r + 1 d is a complex function of position since its differential + idd is a multiple (-) of dr+ird0 of the first factor of cZr 2 + r 2 d! 2 ; and log r -hid is a function of 60 GEODESICS IN TWO-WAY SPACE Just as the position of any point in the plane is given by means of the complex variable x + iy, so the position of any point oh a surface is given by means of the complex variable u where u is an integral of A(0) = 0. (45.2) 46. Conjugate Harmonic Functions. Mapping on a plane. Let,tts now consider the equation A, (0)^0, (46.1) that is, -~i0 l + 7 td(f>* = 0, (46.2) C Ct'^ C) jC'n where 1 = a u fa + a l *fa, 2 = a l2 fa + a**fa. (46 . 3) The expression a! ^fdx l v& fa dx 2 is thus a perfect differential if A 2 (0) is zero; and we have alp = fa, a*<l> 2 = -fa t (46.4) and therefore cdty 1 fa, a*\p fa, (46.5) It follows that A 2 (fa = 0, A (fa fa = 0. (46 . 6) Thus if <f> is any integral of A 2 (0) = we can by quad- rature find *//, another integral of the equation, and the two curves = constant, ^ = constant will cut orthogonally. A real function, annihilated by the linear operator A 2 of the second order, is said to be a harmonic function. The function >//-, obtained as explained by quadrature from 0, is called the conjugate harmonic function to 0. It will be noticed that the function conjugate to i/r is not <f> but </>. We also have A (0) = A (i/r), (46 . 7) and therefore, since A ((/>, \fr) = 0, we see that A (0 + i\fs) = 0. (46 . 8) The function + i^ is thus a complex function of position on the surface. If we take u = 0, v = y\r we have cfo 2 = (A (0))- 1 (du* + dv 2 ). (46 . 9) Thus the problem o/ mapping any surface on a plane, .so that the map may be a true representation of the surface as CONJUGATE HARMONIC FUNCTIONS .61 regards similarity of small figures in each, just depends on the solution of the equation M0) = 0. ' (46 '.!) The magnifying factor from the surface to the plane is A (<f>). Thus to map any surface, applicable on a sphere of unit radius, and whose ground form may therefore be taken as du 2 4-sin 2 ucfe 2 , (46.10) upon a plane we have A 2 ($) = ; and this tells us that <f> must be a function of , /. u\ logftan-J + iv. \ t / We thus obtain Mercator's Projection (ni ^ tan-), y-v. (46.11) The theory of conjugate functions of position on a surface can be applied to problems in Hydrodynamics and Electricity as has been done in the case of the plane. Thus if is a harmonic function on the surface, we may take it to be the velocity potential in the irrotational motion of a liquid over the surface, and \/r, the conjugate harmonic function, will then be the stream function. Conversely, if the ground form is taken to be ds 2 = e(du 2 + dv*), (46. 12) u and v will be conjugate harmonic functions. CHAPTEE IV TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE 47. A quaternion notation. So far we havo been think- ing of the two-way space associated with the ground form ds* = a ik dx { dx lt ; we must now think of that space as a surface locus in Euclidean space. Let i', t", t" r be three symbols which are to obey the associative law and the following self-consistent laws : , <Y" = -4", iY = - 1, i'Y' = - 1, "Y" = - 1. (47 . 1) Let a/, x" , x 1 " be three ordinary numbers called scalar quantities, then, if x = x' i' + JL" i" + x'" i", (47.2) x may be said to be a complex number. If we take y = y ' L ' + y " L " + y '" /", we see that xy = - (x'y' + x"y" + x'"y'") + (x"y'" - x'" y") i' + (x"'y'-x'y'") i" + (x'y" -x"y') i"', (47 . 3) so that xy consists of two parts, a scalar part and a complex number. We write the scalar part Say or xy, (47 .4) and the complex part Vxy or xy. (47 . 5) It follows that x 2 is a pure scalar. We may easily verify the following results : xy~yx = 2xy, (x + y) 2 = x 2 + y A QUATERNION NOTATION and, by multiplying the two matrices 63 y" 11 W' 10" we verify that tixyzw xwyz xzyiv, Vxyz = zxy yzx, Vxyz -f Vyzx -\-Vzxy-Q. If we take i, i" y i" to be unit vectors in the positive directions along the axes of rectangular Cartesian coordinates, then x will be the vector from the origin to the point whose coordinates are x', x", x"' . The length of the vector x will be denoted by | x |. The symbol xy will denote a vector at right angles to x and y, and in the sense that, if the left hand is along x and the right hand along y, then the direction xy will be from foot to head ; the magnitude of the vector will be \x\ | y| sin 6, where is the angle between x and y from left to right. The scalar xy will be equal to | x \y\ cos 6. 48. Introduction of new fundamental magnitudes and equations. Now let z be a vector whose components z\ z" , z"' are functions of the parameters x l and o? 2 , that is, of the coordinates of the two-way space. We have dz = z p dx p and The vector z traces out a surface. Let the unit vector drawn at the extremity of z normal to this surface be denoted by A. We have proved [in 13] that z.^ is parallel to A. We therefore have 3 .^ = /2^A, where fl^ is a scalar quantity. We know that an.d therefore . (48.1) (48.2) 64 TWOWAY SPACE AS A LOCUS IN EUCLIDEAN SPACE Multiplying across by X and taking the scalar product we have, since \\ h - , XX 7 , = 0, \z t = 0, the equation fl ik . h = fl ih . % . (48.3) This is true for all values of i, h, k from 1 to 2 inclusive, and flfa = /2fc$, so that -On-2 = ^i2-i ^ 2 . 1= /2 12 . 2 . (48.4) These equations are known as Codazzi's equations. 49. Connexion of the magnitudes with curvature. The length of the perpendicular from a point at the extremity of the vector z + 8z (where 8z is not necessarily small) on the tangent plane at the extremity of z is -\8z. (49 -1) If we now take Sz so small that cubes of 8x^ 8x% may be neglected, the length becomes that is, iC^n^i + 2f2 12 8x 1 8x 2 + fi, 2(2 $ x *)' (49 . 2) The radius of curvature of any normal section of the surface is therefore given by * ^*^i^ (49. 3) r\ il fl * ill' * ' j-i/ He, *^t / in the tensor notation, and the principal radii of curvature are consequently given by l "-"' ^~^ 2 =0. (49.4) The product of the reciprocals of the principal radii of curvature is therefore /2 n /2 22 -/2f 2 (49 g) Now we saw that (1212) = ^.^^.^-^-ii -^' = /2 n /2 22 -/2; a , (49 ..6) and therefore the invariant K is just the measure of curvature. CONNEXION OF THE MAGNITUDES WITH CURVATURE 05 We thus have the equations Ka = n n nn-ni*> .(49-7) n a . t = n u . lt A w -i-=A 22 > (48.4) wherewith wo are to determine the functions n u ,n M ,n M * (49.8) When wo have found these functions we can find the principal radii of curvature by aid of the equation /2 w -/2J a = 0, (49.9) which may be written i - ^n^+a-^n^^-n-,) = o, (49. 10) applying the tensor notation to the coefficient of -^ If wo were to keep strictly to the tensor notation we should write /2 n ,f2 22 /2| a as /2. We must distinguish between the integer which denotes merely a power, as in /2j 2 denoting the square of /2 12 , and the integer which we called the upper integer in a tensor component. The two meanings are not likely to cause any practical difficulty in reality. SO. The normal vector determinate when the functions f2jj c are known. We must now show how we may determine the unit vector X when the functions n ik are known. * [It is usual to speak of the functions fl u , n 12 , n 22 , i.e. (by 50) ZjAj, SjAj = -2^i> ~a^2 as tho fundamental magnitudes of the second order, those of the first order being the a u , a n , a.,^ or c, /, g of the ground form ds* y and to say that tho six are connected by Gauss's equation (49 . 7), in which K ( 43) is a known function of the magnitudes of tho first order and their derivatives, and by the two Codazzi equations (48.4). Written at greater length these two equations are sa -{i2i}n la = - n ja -{222}n ja -{22i}n ll , v x^ f- %i and their explicit forms are obtained by substituting in these for {HI}, &c., from 43.] 2843 K 66 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE We denote the ground form of the spherical image, that is, of the sphere traced out by a unit vector drawn through the origin, parallel to the normal at the extremity of z, by a'ij.tL'idXk, (50.1) so that 0,'^ = A t -A 7r (50 . 2) , If X . ik =\ ik -{ikt}'X t , where {ilct}' refers to the ground form of the spherical image, we see as before that X. iJc is parallel to the normal to the sphere at the extremity of A : that is \. ik is parallel to A. Now AA t - is zero, and differentiating we have M ik + \i\ k = > so that A A . ik + Ay A/. = 0. It follows that \. ik = A;A 7( .A = -u' tV .A ; (50 . 3) and as we have shown [in 30] how, when ' t -/ f ... are given, A can be obtained by aid of Biccati's equation, we have only to show how a' ik ... can bo expressed in terms of a ik ... and Along a line of curvature we have cfe+RdA = 0; (50.4) let R f and R" be the principal radii of curvature, and let us choose the lines of curvature so that they may be the para- metric lines, that corresponding to R' being efajj = 0, and that corresponding to R" being da^ = 0. We therefore have ^H-jR'Aj = 0, z 2 + R"\ 2 = 0. (50. 5) And it follows that a n = R'f2 n , c* 12 = 7J'/2 12 , a 12 =ft"/2 I2l a 22 = /T/2 22 , /2 U = fi'a' n , /2 12 =.J2'a' M , /2 12 = JZ"ce' 12 , /2 W = E"a' i2f (50.6) THE NORMAL VECTOR TO THE TWO-WAY SPACE 67 so that a n -(J8 / + J B // )/2 11 + a / 11 B'JfZ / '= 0, fl n + a r Vi ll'R" = 0, tln + a'nR'&" = 0. (50.7) Now the expressions on the left in these equations are tensor components, and therefore, as they vanish for one particular coordinate system, they vanish for all systems. That is, the equations are identities. We may express the identity in the form dz* + (K + R") dzd\ + R'R"d\* = 0. (50 . 8) We thus see how o/^... are obtained. We see that n ik = ^ = z^ (50 . 9) for \z i = 0, and therefore ^ z ik + ^u z i > which gives ^ z 'ik + ^k z i 0- From the equations Aj^ = /2 n , Ajj?,, = ^ = /2 12 , Ajjj 2 = /2 22 , A^j = 0, Xz^ = (50.10) we can find Sj and 2 when A is known, and thus determine z by quadrature. We have now shown how the determination of the surfaces applicable to the ground form depends on the determination of the functions fl^ . But here comes the difficulty : the equations to determine these functions flu-2 = ^la-n Au-i = ^12 2' K* ^ fliA-flfa are differential equations of the second order which, in general, we cannot solve. In one very special case we can solve them, viz. when the invariant K is zero. In this case we have shown that the ground form may be taken to be (50.11) 68 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE The equations now become - - u-wio -~ C u*i i , ~ "*|o "~~ C " *'2'2 ^ 12 c^ 2 U ZX> 2 ' 2 ^ 22 and therefore , where - (50 12) 2 ~ (J - ; We can easily prove that we are now led to developable surfaces. 51. Eeference to lines of curvature. The measure of curvature. When we refer to linos of curvature as para- metric lines we have, in (50 . 6), a la = ,R'/2 12 , a 12 = J2"/2 12> and therefore, unless R' and R" are equal, we must have a la = .0,2 = 0. ^ < 51 - 1 .) If the radii of curvature are equal, operating with I and 2 which annihilate a ik , we have 2 . 2 = 0. (51.2) Similarly by operating on a n = -R/2 U , 22 = J?/2 22 , we have li 2 if2 11 + JR/2 11 . 2 = 0, JS 1 /2 22 + ^^22-1 = - ( 51 3 ) From Codazzi's equations we deduce that As wo cannot have /2 n /2 22 /2 25 unless 7J is infinite, we must have jRj = 0, J2 2 = 0, (51 .4) that is, U is constant and the surface must bo a sphere. Leaving aside the special case of a sphere, we have when the parametric lines are the lines of curvature a 12 = /2 12 = a' 12 = 0, (51 .5) REFERENCE TO LINES OF CURVATURE 69 and wo can often simplify proofs of theorems by referring to lines of curvature as parametric lines. The vector z^z 2 is clearly normal to the surface at the extremity of z : its magnitude is ai (or h as it is generally written) and therefore z^ 2 = a^A. Similarly we have ^1^2 ^ tt '^ The expression Kz i z k ~X i \j c is a tensor component. It obviously vanishes when we refer to lines of curvature: it therefore vanishes identically and WehaVO K^z 2 = \X 2 . (51.6) We then have Kal = a'*, (51 . 7) that is, the measure of curvature is the ratio of a small element of area on the spherical image to the corresponding area on the surface. 62. Tangential equations. Minimal surfaces. We shall now develop some further formulae. We have /2/2,-/2? 2 =aK = a'K~ l = (aa'; n* = + jn. (52.1) and, from the formulae connecting a ik> u 'ik> ^ihi we easily deduce a' ik fl ik = R' + R", We can also obtain formulae applicable to a surface given by its tangential equation. This means that instead of beginning with a vector z, given in terms of parameters x l and aJ 2 , we begin with assuming that X is known in terms of these parameters, and also p, the perpendicular from the origin on the tangent plane to the surface. The lines of curvature are given by = 0, = 0. 70 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE They are therefore also given by (n, n -Ra\ l )dx l -}-(n^~Rii' v ^dx^ = 0, (/2 ia - JRu' 12 ) c/^-f (/2 22 - JKa' 2a ) tte a = 0, (52 . 3) as we see at once from the connecting equations. The tangential equation of a surface is p + Xz = 0. (52.4) By differentiation wo deduce that With reference to the ground form of the spherical image we therefore have p . ^ + x . ik z + f2 ih = 0. Now A.^ = -a'. 7 ,A, and therefore />. ^ + a'^> -f- fl^. = 0. ^ (52.5) When therefore we are given the tangential equation of a surface, the lines of curvature and the radii of curvature are given by the formulae i = 0, (52.6) In particular if we want the parametric lines to be lines of curvature on the surface we must have and therefore > must satisfy the equation. *>.,= 0. (52.7) There is a particular type of surface with which we shall have to do : the minimal surface characterized by the property that the principal radii of curvature are equal and opposite. The expression SAA^-SAA^ (52 . 8) is a tensor component.. If the surface is a minimal one .it vanishes when we refer to lines of curvature, and therefore TANGENTIAL EQUATIONS. MINIMAL SURFACES 71 it vanishes always if, and, we see, only if, the surface is a minimal one. We always have the formula, as we easily see, The tangential equation of a minimal surface is therefore given by A'.p + 2^> = 0. (52 . 9) If we refer to the null linos of the spherical image as para- metric lines, the ground form of the sphere becomes 4 (1 + x l (i' 2 )- 2 dx l dx. t i and the equation which p has to satisfy becomes It may be shown by Laplace's method that the most general solution of this equation is (l+x^p = 2x l f(x 1 )+2,*\ 2 <f) (x 2 ) + (1 +*- 1 ag (x\f (x^ + xl f (0 2 )), (52 . 10) and we have thus obtained the tangential equation of the minimal surface. 63. Weingarten or W surfaces. We now proceed to consider more generally surfaces which, like the minimal surface, are characterized by the property that their radii of curvature are functionally connected. These surfaces are called W surfaces, after Weingarten, who studied their pro- perties. When we refer to the lines of curvature as parametric lines we have (50. 5) ^ + 1^ = 0, c 2 + #"A 2 ^0, and therefore (R'-R") A 12 = JK''^- JR'.^. Let R"=f(R r ) n dx and (x) = ej *-/(*) . We easily verify that f (x) $ (x} <f>"(x) 1. 0'fa) ' - ' { ' 72 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE Now . R'.\ _ /' (K) R\ _ 0" (R>) R\ a .'-." ~ R r -t(R'} ~ "~0'TT~ ~ ^ g '^ ( ^' *' - JT - If - The equation satisfied by A thus becomes X v=\ 4 lo o' (*' W) - \ ^ log (^ (JZO), (53.2) and therefore, since A,X 2 is zero, We may therefore, the lines of curvature still remaining the parametric lines, take +l=0. (53.3) The spherical image of the TT surface (that is, it will bo remembered, the surface traced out by a unit vector parallel to the normal at the extremity of the vector z, and expressed in the coordinates which give z), when the W surface is referred to the lines of curvature as parametric lines, will be therefore 2 It will be sometimes more convenient to express the para- metric coordinates by u and v. Conversely, if we are given the ground form of a sphere in the form pdu? + qdv*, where p and q are functionally con- nected, it will be the spherical image of a W surface referred to its lines of curvature. 54. An example of W surfaces. We may now consider some examples. We saw ( 42) that, referred to what we called focal coordinates, the ground form of any surface may be taken as J 8 2 _ Sec 2 $ dv * + coscc a 0^ (&*.l) where 2u = PA + PB, 2v = PA- PB, and A and B are any two points on the surface which we call the foci ; PA and P$ are geodesic distances and 2 6 is the angle APB. AN EXAMPLE OF W SURFACES 73 If the surface is applicable on a sphere we see that . sin(c v)sin (c + v) tan C7 = ".- , ------ r : ---- : c) where 2e is the geodesic distance between A and B. Thus (#') = cos 6, <$>' (R') = cosec (9, and therefore cosec 6 ^ = sin d. civ If we now integrate this equation we- have R' = sin 2 0-20 4- e, where e is some constant. But JK'-jR" = ^S- = sin 6 cos (9, (/> (R) so that 4E" = -20- sin 20 + e, and therefore 2 (R'-R") = sin (e - 2 E' - 2 .K"). (54 . 2) This is the relation between the principal radii of curvature of the W surface which corresponds to the spherical image sec 2 6du* + cosec 2 6dv 2 . (54 . 1) In this case we know the radii of curvature in terms of the parameters since 6 is so known. We thus know the ground form both of the surface and of the spherical image, and there- fore can find the surface as a locus in space. 55. The spherical and pseudo-spherical examples. In the above example we began with a known ground form for the spherical image and deduced the relation between the curvatures. If we take any knowft ground form for the spherical image 2*lu 2 -f qdv 2 , where p and q are functionally related, and known in terms of the parameters, we could proceed similarly. We could find the relation between the curvatures and we should obtain in known terms of the parameters the ground form of the. surface. We could then obtain the surface as a locus in space. In my exposition of the method I have followed 74 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE Darboux and taken the example he gives, as I #lso do in what follows. When on the other hand we begin with a known relation between the curvatures, we cannot in general find the surface as a locus in space. Thus, let us &pply the method to the problem of finding the surfaces applicable on a sphere of unit radius. Here we have R' R" = 1, (55. 1) and we may take R' = co th 0, R' 1 = tanh 6. The function which expresses R" in terms of R' is p dR' PT T and (f) (R') = e R' = cosech 0, </>' (R f ) = cosh 0. The ground form of the spherical imago is'thus sinh 2 (9 c/u 2 + cosh 2 6dv*. (55 . 2) On the sphere the measure of curvature is unity, and therefore our formula for K gives 6 n + #22 + inh cosh 6 = 0. (55 . 3) Now if we knew how to solve this equation we should have an expression for in terms of the parameters u and v, and we should thus be able to write down the ground forms of the surface and of the spherical image in terms of the parameters ; and thus have the means of determining as loci in space all the surfaces which are applicable on the sphere. Unfortunately wo cannot solve the equation generally. This example shows how ultimately nearly all questions in Differential Geometry come to getting a differential equation ; and that the complete answer depends on the solution of the equation. But even when wo cannot solve the equation we gain in knowledge by having the differential equation in explicit form. Thus it happens sometimes that two apparently quite different geometrical problems may depend on the sqme insoluble differential equation. The surfaces connected with THE SPHERICAL AND PSEUDO -SPHERICAL EXAMPLES 75 the problems are thus brought into relationship with one another; and the relationship ia sometimes very simple and very beautiful. Illustrations of this will occur later. All we can say now is that the differential equation n 4- 22 + sinh 6 cosh = is that on which depends the obtaining of all surfaces which are applicable on the sphere : that is, the surfaces whose geodesic geometry may be considered as absolutely known, being just spherical trigonometry. Similarly we might consider the problem of finding the surfaces applicable on a pseudosphere. Here we have JZ'li" = -l, (55.4) and wo take R' = cot 0, R" - tan 0. We find that <f> (cot 0) coscc 0, 0' (cot 6) cos 0, so that the ground form of the spherical image is sin 2 0Ju 2 + cos 2 0(fo 2 , (55.5) and the equation to determine 6 is 22 -0 u -fsin0cos0 = 0. (55.6) If we apply the substitution 2ti'=w + v, 2v' = u-r, 20 = 0', the equation takes the simpler form 12 = sin0; (55.7) and on this equation depends the obtaining of the surfaces with the known pseudospherical trigonometry, obtainable from spherical trigonometry by writing ia, ib s ic, for the arcs of a spherical triangle. 66. Keference to asymptotic lines. We have now con- sidered the surface when referred to lines of curvature as parametric coordinates, and the equations resulting, z l = U'Aj, 2 = R"\> where R' and R" are the principal rajlii of curvature and A is the unit vector parallel to the normal at the extremity of z. 76 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE We now proceed to consider another special system ot coordinates. ' The 'elements dz and z on the surface which are drawn through the extremity of the vector z are perpendicular if dz 8z = 0; that is, if cl </#! Sx l -f ^ c 2 (dx l 8x^ + dx^Sx^ + z\ dx 2 Sx 2 = 0, or a ll dx l 8x l + a l2 (dx i 8x t2 + tlxi8x 1 )+a, i <>dx 2 $x t , = 0. (56.1) The elements dz and Sz at the extremity of z are said to be conjugate, if the tangent planes at the extremity of z and at the extremity of z + dz both contain the element 8z\ that is, if Sz is perpendicular to the normals at the extremities of z and of z + dz. We therefore have for conjugate elements 8zd\ = 0, that is, z l \ l dx l Sx 1 + z l \ 2 (d i f\8x 2 -{-(Jx 2 8.r l ) + ^ 2 \^dx 2 8a\,, 0, or /2 ll rfo' 1 r 1 + n ]t> (djf' l 8x^dx^S^ l ) + /2 2 .//.f 2 <Stf, 2 = 0. (5G.2) Thus we see that the lines of curvature at any point of a surface are both orthogonal and conjugate, and conversely we see that lines which at any point are both orthogonal and conjugate are lines of curvature. An element which is conjugate to itself satisfies the equation fi ll dxl+2fi n dx l dx 2 + ftM(fa% 0. The self-conjugate elements at a point form the asymptotic lines n n dx\ + 2 fl^lx^a^ fl^(lx\ = ; (56 . 3) and we see that the radius of curvature of a normal section in the direction of an asymptotic line is infinite. fix ft f We call ~ and -.- a the 'direction cosines' of an element ds ds on the surface. They tell us the direction but they are not the cosines of the angles the element makes with the para- metric lines. We often write them in the tensor notation I 1 , 2 ; but we must remember - is not the square of , nor REFERENCE TO ASYMPTOTIC LINES 77 is Qf 1 ) 2 a tensor component 12 , but the square of l . We have identically ^'i-^i k = 1 ; and, if R is the radius of curvature of any normal section of the surface, Rfyk?? = 1- ( 5G 4 ) Take now the asymptotic lines as parametric lines. We have n n = o, n 2 , = o, and therefore by Codazzi's equations /2 u /2. 2 -/2 J2 /2 12 -7va = l we have log/2 12 = (111 j- {212], Now we saw ( G) that the determinant a satisfied the equations :\ -- o y and therefore c - (log 7v r ^) -f 2 { 2 1 2 } =0, ^ (log /vi) -f2{121} = 0. (56.5) These arc the equations which tho coefficients a^ must satisfy if the parametric lines are to be asymptotic. If we are given any ground form, and if we could transform it so that the new coefficients would satisfy the above equations, then we could, since in this case we would know the functions /2 U , /2 r> , /2 20 and the ground form, find the surfaces to which the form would be applicable. But the transformation would itself involve the solution of differential equations of as great difficulty as Codazzi's equations. Taking the asymptotic lines as coordinate axes we have ^A! = 0, z 8 A 2 = 0, 78 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE and therefore z l = ^>AA 1 , (56 . 6) where p is some scalar. Similarly we have ^ _ q ^ (56 . 7) where q is a scalar. As 3X = A we have 2)S\X 1 X 2 = qS\\ 2 \ l = and therefore ^ ~ ~ 7- (56.8) Since A! A., = Kz^z^ we have - 7^ 2 FAA T AA 2 = A^s , that is, -7i^ 2 (A 2 /S t AX 1 X-A&'XA l X 2 ) = C A 2^ or A> 2 XAS r A'^ = X, (56.9) since A X A 2 is parallel to A. We therefore have p ( K )~t, (56.10) and s 1 =(-A r )"*XX l> 2 - -(-A r )-*XX a . (56.11) These are the exceedingly important equations which we have when we choose the asymptotic lines to be the para- metric lines. 67. Equations determining a surface. If we now take Z=(-K)~*\, (57.1) so that Z is a vector, parallel to the normal at the extremity of z y and of length ( A r )"J, we can write the equations which determine the surface in the simple form From these equations we have ^12 = 0, and therefore Z 12 = pZ, (57.2) where p is some scalar [not the p of (56 . 10)]. In order to find the asymptotic lines of a given surface we have to solve the ordinary differential equation v* = 0, EQUATIONS DETERMINING A SURFACE 79 and when we have done this we can bring the equation of the surface to the form stated. We have Z = cA, '(57.1)' and we notice that c is an absolute invariant. Differentiating we see that and therefore cA^ = CupCy that is, ca' 12 = C 12 pc. (57.3) From the formulae a' 12 = fl^f-^, + ITT/ )> we see that and therefore p = - /2 12 ^ -f -^ - (57 . 4) The equation of the surface referred to the asymptotic lines is therefore z l = ZZ 19 2 = -ZZ^ (57 . 5) where Z n = -/2 W + ^. (57.6) 68. The equation for the normal vector in tensor form. We can express the equation which the vector Z must satisfy in tensor form so as to be independent of any particular coordinate system. The null lines on the surface applicable on the ground form are the lines which satisfy the equation a ik dx { dx k = 0. On a real surface they are of course imaginary and are characterized by the property that the distance, measured along the curve, between any two points on a null curve is zero. Let us now consider the ground form fl ih d'^dx^ ' (58.1) 80 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE remembering that any quadratic differential expression is the ground form of some set of surfaces. The surface, to which this form applies, will have as its null lines the correspondents of the asymptotic lines on the surface we are considering. Let Beltrami's differential operator with reference to the ground form fl^dx^lx^ be denoted by <yA 2 . (58 . 2) Now we saw (43 . 24) that, with reference to the null lines as parametric lines, that is with reference to the asymptotic lines on the surface we are considering, 2. /2 12 du dv The equation Z n = may be written that is, tt A 2 ^ = (_ 2 ( + - /7 ))^ ; (58.3) and this is a tensor equation independent of any coordinate system. 59. Introduction of a new vector We may write this tensor equation briefly in the form ^Z = pZ. (59.1) Let be any scalar quantity which satisfies the equation A a = jp0. We then have \ K = Z A 2 6. Now we saw (17.4) in the chapter on tensors that where ^u f and therefore u t v l = v t u*. INTRODUCTION OF A NEW VECTOK 81 We may then write the equation 6A.,Z-Z\0 ^ in the form 6 * VttZ<< = Z -- Z6 ( ) Q (59.2) [where /2 denotes /2 n /2 22 ~/2i 2 ]. If the asymptotic lines are real /2 will bo negative : we therefore write this equation , .- __. c , -!#.{__ #02j_ Ot (59.3) We can then by quadrature find a vector such that that is, V-/2 = 7(/2 1 ^ 1 -/2 ]1 ^)-tf(/2 1 ^ 1 -/2 ll ^), ^-^,,^,)-^^^,-^^). (59.4) It should ho noticed that to find required a solution of the equation ^ g _ ^ e ( 59 B 5 ) 60. Orthogonally corresponding surfaces. We have V(A,,Z)Z= 0, and therefore or, since ^72/^ = 0. (GO . We can therefore by quadrature find a vector z such that z l = V that is, ^ y ^-/2 I2 ^ 2 ). (60.2) If the parametric lines are asymptotic these arc just the equations we began with. 2843 T\f 82 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE We see at once that 3i = 0, 3j/ 2 + 2 = 0, 0^ 2 = 0, and therefore corresponding elements of the surfaces traced out by z and by are connected by the equation dsdf=0 t (60.3) that is, corresponding elements are perpendicular to one another. The surfaces are then said to correspond orthogonally to one another. 61. Recapitulation. We may now restate the results we have arrived at. Consider the ground form and let A 2 have reference to this form. Let Z be a vector which satisfies the equation Z) = 0. Then z l = V, 2 define a surface traced out by a vector z. On this surface the unit vector parallel to the normal at the extremity of z is given by #=cA f where c ( K)~* 9 and K is the measure of curvature of the surface z. We have ^Z = Z . A.,c where 2 , = __ The asymptotic lines on z are n ik dxt The surfaces given by where 6 is any scalar satisfying the equation . **6=p0 9 correspond orthogonally to the surface z. RELATIONSHIP OJ? SURFACES Z AND 83 62. Relationship of surfaces z and When the para- metric lines are asymptotic on 0, that is, when n n = 0, /2 22 = 0, and therefore f M = |* f 2 + -^ . (62 . 1) The parametric lines on are now conjugate lines : for if we have an equation of the form where p and q are any scalars, Xj l2 = 0. If p 2 = ? l the conjugate lines have equal invariants in Laplace's sense. The parametric lines on are therefore said to be conjugate lines with equal invariants. To the asymptotic lines on z there correspond therefore conjugate lines with equal in- variants on g. If on any surface we are given the conjugate lines with equal invariants, we can find by mere quadrature a surface which will correspond orthogonally to . For if ~ ~ where 6<f> = \ and therefore ~- (<f> 2 ^) + ~ (<^) = o, (62 . 2) <j 16 o V where <j> 2 means the square of (f> and is not a tensor notation. We can therefore find by quadrature a vector Z such that that is, f l = The surface given by 84 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE will correspond orthogonally to and will have the asymptotic lines as parametric lines. We have now seen the relationship to one another of the surfaces z and and the method by which, given either, we are to obtain the other. 63. Association of two other surfaces with a c-surface. Let a vector m be defined by the equation We have, taking as parametric lines the conjugate lines with equal invariants on and therefore O^Z-m) = Otfi + mJ, 6. 2 (Z+m) = 0(# 2 -wJ. (63.1) From these equations we see that Z 12 Z o, m J2 m = o, and, as V(Z-m)(Z l + m 1 ) = 0, V(Z+m)(Z.,-m^ = 0, ZZ l luiihi -f ZiUi 4- ^^^ = 0, J&ziMi z Zm 2 Zs>ti = 0. (G3 , 2) We can take s, = ^, , z. 2 = ZX^ and we have 2/i ~^i T~" It follows that y only differs by a constant vector from s + Zni. We have thus obtained the surface y, where 2/ = s-{-#m, (63.3) directly from z and and the asymptotic lines on this surface correspond to the asymptotic lines on z. FOUR RELATED SURFACES 85 64. \Vc obtain yet another surface directly from the definition ?; = #0, ( 64 -J) where 6(/> = 1 ; and we see that so that 77 L m(f> l f in l (f>. (04 . 2) Similarly we see that rj^ m 2 (j)~ m0 2 . (G4 .3) The surface 77 will therefore correspond orthogonally to the surface y ; and to the asymptotic lines on y will correspond on 77 conjugate lines with equal invariants. We have thus four mutually related surfaces, s> 2/> ^ which are intimately connected with two problems in the Theory of Surfaces, viz. the theory of the deformation of a surface, and a particular class of congruences of straight lines. The relations between the four surfaces will be more com- pletely stated when eight other surfaces are introduced, as they will be when we consider the Deformation Theory. CHAPTER V DEFORMATION OF A SURFACE, AND CONGRUENCES 65. Continuous deformation of a surface. We have seen that the problem of determining the surfaces in Euclidean space, to which a given ground form appertains, depends on the solution of the equations and we have pointed out the difficulty of solving these differential equations. There is a related problem the solution of which is simpler. This problem is the determination of a surface differing infinitesimally from a given surface and applicable upon the given surface. Let z be the vector of the given surface, and z + t the vector which describes the neighbouring surface which we are seeking, t being a small constant. We may regard t as a small interval of time and ^ as a linear velocity vector, descriptive of the rate of increase of z, as we pass to the neighbouring surface which is applicable upon the given surface ; or as the growth of the vector z under the condition of preserving unaltered the element of length. If we can obtain we have the vector which defines the continuous deformation of a surface. We have at once i& = 0; X& + * a fi = 0; * 2 &=0, (65.1) CONTINUOUS DEFORMATION OF A SURFACE 87 that is, the vector describes a surface corresponding ortho- gonally with the given surface described by z. An interesting and immediately verifiable theorem on surfaces which correspond orthogonally is the following : ' If z and { correspond orthogonally, then the surfaces traced out by z -h and z are applicable on one another ; and conversely, if z and are the vectors of two surfaces applicable on one another, z + and z will be the vectors of two surfaces which correspond orthogonally/ 66. A vector of rotation. From the kinematical relation of the vectors z and wo see that d is the relative velocity of the extremities of dz in the deformation of the surface z. In the deformed surface the element which corresponds to dz will have the same length but will have turned through an angle. Let the rotation necessary to produce this be represented by the vector tr. Now if a vector a, drawn from a point, is made to rotate with an angular velocity whose magnitude and direction is represented by a vector r, drawn through the same point, the linear velocity of tho extremity of a will be given by ra. It therefore follows that df = rdz, or 1 = ^1; & = ^V (66. 1) The vector r is parallel to the normal to tho surface ^, at the extremity of tho vector g. We therefore have where a is some scalar ; and therefore since c^ = ; ^ 2 + ^ = ; 5j = 0, so that r = ^y; (66.2) and thus r is uniquely obtained, when z and { are known. 88 DEFORMATION OF A SURFACE, AND CONGRUENCES 67. Geometrical relationship of surfaces traced out by certain vectors. In exactly the same way we see that where P=T L ^' ( G7 - 2 ) &Vp2 By differentiation of the equations we sec that r^ l ~ r,^; (G7 . 3) and therefore the vectors r t , r 29 y lt z 2 are all parallel to the same plane. It follows that the normals to the surfaces traced out by z and r are parallel at corresponding points. Similarly wo see that the normals to the surfaces traced out by the vectors p and are parallel at corresponding points. But the vector r is parallel to the normal at the correspond- ing point of : it is therefore parallel to the normal at the corresponding point of p. From the equations r* ^ we see that rp = 1. (67 . 4) It follows that the r and p surfaces are polar reciprocals with respect to a sphere whose centre is at the origin and radius the square root of minus unity. 68. The angular velocity r is applied at the extremity of the vector z. Now an angular velocity r, at the extremity of the vector 0, and an angular velocity r at the origin, are equivalent to a linear velocity zr. It follows that a linear velocity and an angular velocity r, at the extremity of 0, are equivalent in effect to a linear velocity + zr, and an angular velocity r at the origin. We are thus led to consider two other vectors, +zr and TWELVE ASSOCIATED SURFACES 89 69. A group of operators, and a system of twelve associated surfaces traced out by vectors. The fundamental relations between the vectors 0, r, p, arc expressed by the equations d{=rds; dz = P 7t{. (69.1) These relations are unaltered by the transformation scheme in 0, r, p, *' = ^ + P; f = r; r' = ^ ; p' = s, (09 . 2) which we shall denote by the operator A. They are also unaltered by the transformation scheme ' = {; f=; r' = p; p' = r, (69.3) which we shall denote by the operator S. We see that the operators A z , A*, A 4 , A'' tiro respectively the transformation schemes . /' _ _ P. . We see that A 6 = 1 ; B* = I, (69 . 4) and A*B = BA; A*B = BA 2 ; A*B = BA* A 2 B = BA* ; AB = BA*, and so the operators A and B form a group of order twelve. The operators A form a sub-group of order six ; -the opera- tors /{ form a sub-group of order two. If we take p = A 3 Q = BA ; R = A 2 we have P 2 = 1 ; Q 2 = 1 ; R 3 = 1, (69.5) PC = QP; PR = RP;QR = R*Q -, QW = RQ, and the operators P, Q, R will generate the same group. Of this group the operators P form one sub-group, the operators 90 DEFORMATION OF A SURFACE, AND CONGRUENCES Q another sub-group, and the operators P and Q together a sub-group of order four. The operators R form a sub-group of order three. We thus obtain directly from the four vectors 0, r, p a system of twelve vectors which trace out twelve surfaces connected in various ways at corresponding points. 70. We may arrange the twelve surfaces in tabular form thus The first column will denote a vector of a surface; the second the vector of the surface which corresponds ortho- gonally to the surface in the first column and in the same row; the third column will denote the vector which gives the angular velocity corresponding to the surface in the same row but in the first column; the fourth will denote the angular velocity which corresponds to the surface in the same row but in the second column. The vectors in the third column are parallel to the normals to the surfaces in the second column and in the same row ; the vectors in the fourth column are parallel to the normals to the surfaces in the first column and in the 8ame row. Finally the surfaces in. the same rows and in the third and fourth columns respectively are reciprocal to one another. THE TWELVE SURFACES FOKM THEEE CLASSES OF FOUE 91 71. The twelve surfaces form three classes of four. Let us now recall what wo proved about the four surfaces which we denoted in 62-4 by 2, , y, 77, and the equations of con- nexion when z is referred to its asymptotic lines. We had mO, Z = 77$, y = z -f Zm. Wo see that Z is parallel to the normal at the extremity of z, and p is parallel to the same normal. Therefore Z = pp, (71.1) where p is some scalar. Now z= = O'Z' but Si = ^i, and therefore Z dp, that is, 77 = p. (71 . 2) Now y = z -f Zm z + rjg, and therefore y z -f^p. (71 . 3) The four surfaces are therefore in the present notation (merely changing the sign of the vector 77) that is, s, Hz, BAz, A 5 z, or c, PQRz, Qz, PJlz. Now the asymptotic lines correspond on two surfaces which are polar reciprocal to one another, since conjugate lines reciprocate into conjugate linos; and we know that the asymptotic lines correspond on z and z -h p. The asymptotic lines therefore correspond on that is, on z, Pz 9 Qz, PQz. 92 DEFORMATION OF A SURFACE, AND CONGRUENCES The surfaces which correspond to these orthogonally are respectively ^ ^ + - that is, PQRz, QRz, PRz, Rz. On these surfaces there correspond to the asymptotic lines conjugate lines with equal invariants. We Avill say con- jugate lines with equal point invariants. The surfaces which are respectively reciprocal to these four are , r t ^ f+rz rf { + '' * V' that is, R 2 z, PR*z, QR*z, PQR 2 z. We say that on these surfaces there correspond, to the conjugate lines with equal invariants on their reciprocals, conjugate lines with equal tangential invariants. The twelve surfaces thus fall into three classes: viz. those on which the asymptotic lines correspond ; those on which conjugate lines with equal invariants correspond ; those on which conjugate lines with equal tangential invariants corre- spond. The surfaces of any class are permuted amongst themselves by the operations of the sub-group i; P-, Q; PQ. 72. A case in which one surface is minimal. If the vector z is of constant length we can prove that the surface + w (72.1) is a minimal surface. We saw that the normals at corresponding points of z and of r were parallel. If then z is of constant length, the vector is parallel to its own normal and therefore equal to AA, where k is a constant, and A is the unit vector parallel to the normal at the extremity of r. But and therefore A CASE IN WHICH ONE SURFACE IS MINIMAL 93 We saw (52 . 8) that the condition that z might be a minimal surface was # 01 AA 2 = Uz 2 \\ l9 and clearly this condition will remain the same if we replace X by any vector parallel to it. Let y = g+zr. We see by tho table that z is parallel to the normal at the extremity of y. The condition that y may be a minimal surface is then Sy^zz^ = Sy^zz^ (72 . 2) But by the fundamental formula of connexion and by the table we see that ^ ^ 2/i = - r i 2/2 = 7 V Tho surface y will therefore be a minimal surface if Lliat IS, it ZZ^ZT-i Z T-tZt) ZZ-tZ^^ Z ^Vj^i Now z being of constant length this condition becomes At /" r> . M^2 'v^i and this we have seen is true. This theorem will bo used in proving an interesting theorem of Ribaucour's in connexion with a particular class of con- gruences. We now proceed to consider the theory of congruences of straight lines in connexion with which the twelve surfaces will be of interest. 73. Congruences of straight lines. If we wish to con- sider not merely the geometry on one particular surface but the relation of points on that surface to corresponding points on another surface, we are led naturally to consider the congruence of straight lines which join the corresponding points. Let z be a vector depending on two parameters u and v, and ft a unit vector depending on the same two parameters, and drawn through the extremity of z. Let iv be a length taken along the vector p, ; the congruence will, then be defined by (73.1) 94 DEFORMATION OF A SURFACE, AND CONGRUENCES We regard z and p as functions of the parameters u and v, and therefore the current vector z 1 will bo a function of the three parameters u, v, and w. The unit vector p will trace out a sphere which we call the spherical image of the congruence. Let so that dcr* = a u dfu a + 2a l2 dudv + a 22 dv* (73 . 2) is the ground form of the spherical image. Let Wk = <*ik> ll> ^n a 2U ""^12 and notice that in general >ik = *>li- If we take two neighbouring rays of the congruence we have dz' dz + wd/j. -f p. du\ If X is a unit vector perpendicular to p, and cZ//, da\ = pit //, and therefore Jul<r\ It follows that <o n dv) (a l2 But, if 5 is the shortest distance between the two neighbour- ing lines, 5 -AcZc, and therefore (73.3) a T T du + a . 9 r /?;, a, cZu 4 a 99 dv 11 ' 1 / 7 14 Z^ 74. Focal planes* and focal points of a ray. The value of w which corresponds to the shortest distance between two FOCAL PLANES AND FOCAL I'OINTS OF A RAY 95 neighbouring rays ivS given by the fact that dz' is perpendicular to \L and fi -f d/jL ; and therefore dz'dp. = 0. We thus have dzdp + wdfi 2 = 0, o) u du 2 -f (o) 12 -f eO dudv 4- co^dv 2 " ~ll - ~ - - -- ,, , so that ( 74 .1) ' The critical values of w, say w' and tc", as we vary the ratio du : dv, are therefore given by co n wet -f ox,,) wu - 0, (74 . 2) and the corresponding values of the ratio du: dv are given by = 0. (74.3) There are, by (73. 3), two values of the ratio du:dv which make 5=0. Through each ray of the congruence there thus pass two developable surfaces defined by = 0. (74.4) The planes which ])ass through this ray and touch the developablcs are called the focal pldiies of the ray. The points where the ray is intersected by these neighbouring rays are called the focal points of the ray. The developables are defined by = p (a^d ^du + where p is some multiplier; and we see that this multiplier is w. The focal distances, /' and /", are therefore the values of w which satisfy the equation (74.5) 06 DEFORMATION OF A SURFACE, AND CONGRUENCES 75. Limiting points. The Hamiltonian equation. Prin- cipal planes. If we have any two real quadratic forms we can, by a real transformation, bring them to such a form that in the new variables It is therefore possible by a real transformation to make 12 = 0. (75.1) The points on the ray given by w', w" are called the limiting points of the ray. These points are therefore real. If we suppose the transformation applied which makes we have o> n = w'<t UJ o> 22 = iu"a 22 ; and the value of w which corresponds to the shortest distance between two neighbouring rays is given by ?(/(?!, du' + iv"a 29i dv 2 We may take COS 2 e - j^Li^ 2 S m 2 e = -A 2 -^ 2 ,,, a n du* + <t 2< ,<lv* (( n du* + ^^dv L and we have the Hamiltonian equation w = ti;' cos 2 + ^ r/ sin 2 (9, (75 . 2) showing that the shortest distance between any two neigh- bouring rays lies between the two limiting points. The values of the ratio duidv which correspond to the limiting points are given by (w'-w")dudv= 0. Leaving aside the special congruence when the limiting points may coincide, we see that corresponding to the limiting point w\ du is zero, and the shortest distance is parallel to fjL/ji 2 . Similarly the shortest distance corresponding to to" is PRINCIPAL PLANES 7 parallel to /i/^; and these shortest distances are perpendicular to one another since ^>/*iW*a = W*Wi-PlW* = - ( 75 3 ) The planes through the ray /z which are perpendicular to these shortest distances are called the principal planes of the ray : and they are perpendicular to one another. 70. Principal surfaces, and the central surface. Return- ing now to general coordinates we see that and therefore, in the important class of congruences for which co 12 c 21 , the limiting points and the focal points coincide. We see also that the focal planes will then coincide with the principal planes. When we take any equation connecting the parameters u and v of the congruence we obtain a ruled surface of the congruence. The directrices of the ruled surface will be curves lying on the surface z. If u and v are functions of a variable p, then p and w will bo the coordinates of the ruled surface. The lines of striction on the ruled surface will bo given by ~ cfo8 (76.1) where u and v are connected by the equation which defines the ruled surface. The ruled surfaces given by -f o> 21 ) db\ ^ (co V2 -f o> 21 ) da -f o>^^ v> u + a n dv, a^dvu + a^dv (76.2) are called the principal surfaces of the congruence. The locus of the points on rays midway between the foci, and therefore midway between the limiting points, is called the central surface of the congruence. 98 DEFORMATION OF A SURFACE, AND CONGRUENCES 77. The focal surface. Any ray of the congruence will be intersected by a neighbouring ray if dz + wdp + pdw = 0. The developables which pass through the ray are therefore given by Sdzdpfi = ; that is, by S (z^lu, + z^lu) (fadu + fadv) fi 0. The focal points are given by (z l + W/JL^ du 4- (^ 2 + w/O d v + pd = 5 that is, by S (z^ + W/JL^ (z% 4- w// 2 ) /* = 0. The focal surface of the congruence is defined as the locus of the focal points on the rays of the congruence. If we so choose the parameters that the equation defining the develop- ables is dudv = o, then Kz l jji l p. o, Sz^fajj. = ; so that o 1 = a ^ + 6//, z> c^ + dp, where a, 6, c-, d arc scalars. Substituting in the equation S (z l -f wfij) fa + WfjiJ n = 0, which defines the focal points, we see that the focal surface has two sheets given by Z' = Z (l/JL, Z* = 2C/JL. 78. Bays touch both sheets of the focal surface, The congruence of rays of light. For the first sheet z\ = (I - x ) p, z\ = (c - a) p z + ((Z - ei 2 ) //, so that the normal to the first sheet is parallel to /JL/JL^ ; and the ray touches the first sheet along the u curve on it that is, the curve along which only u varies ; and the v curve is conjugate to the u curve. Similarly wo see that the ray touches the second sheet along the v curve on it, and the u curve on it is conjugate to this. Thus any ray of the congruence touches both sheets of the THE CONGRUENCE OF RAYS OF LIGHT 99 focal surface ; and the tangent planes to the focal surface at the two points of contact are the tangent planes to the developables through the ray. The edges of regression of the developables are the u curves on the first sheet, and the v curves on the second sheet. If the congruence is formed by rays of light, the focal points on the ray are the foci as defined in the theory of thin pencils. F l and F 2 arc the foci on what is called the principal ray of the thin pencil. The tangent plane at f^ to the second sheet, which is the tangent plane at F l to the develop- able, is called the first focal plane : so the tangent piano at F l to the first sheet, which is the tangent plane at jF 2 to the other developable, is called the second focal plane. The devolopables through any ray are somewhat like the above figure. The focal lines as defined in some text-books on Geometrical 100 DEFORMATION OF A SURFACE, AND CONGRUENCES Optics have no meaning at all; but it has been pointed out that the lines conjugate to the principal ray on each sheet have a physical meaning which might entitle them to the name of focal lines.* 79. Refraction of a congruence. Malus's theorem. A congruence is given in terms of the coefficients a^. of its spherical image and of the coefficients o> n , o> 12 , o> 21 , co 22 , We can see how the congruence, when we regard it as formed by rays of light, is altered by re- fraction at any surface z, whose normal is parallel to the unit vector A. Let p be the unit vector into which /z is refracted : that is, let // trace out the new spherical image. We have /*' = f*/z + 6A, where a and b arc scalars. In the ordinary notation of optics, if cf) is the angle of incidence, 0' the angle of refraction, and k the index of refraction, k sin 0' sin 0. Now X/JL' (iXfj., P-'P' bXji; and therefore a sin $ = sin0', 6sin0 = sin (0 0'). (79 . 1) We thus see that a is a constant independent of the angle 0, but b depends on 0. We have A/z-f cos0 0. Since /// = a/j, i + IXj + b { A, we have &>;// where /2^ refers with its usual meaning to the surface of refraction. * [Probably the allusion is to a note 'On focal linos of congruences of rays' : Elliott, Messenger of Mathematics } xxxix, p. 1.] REFRACTION Otf A CONGRUENCE. MALUS's THEOREM 101 We see that cos 0' = a cos + 6, and therefore if we multiply by /JL', that is, by and take the scalar product, we get, since fi'ft'j is zero, (tbiXpi + kjji) = /^cos0'. % (79.2) We notice that if o^., co a then o>' r , fc^r We shall see (83.2) that the condition o> la = o> 21 (79 . 3) means that the rays of the congruence arc normal to a system of surfaces and we now see that this property is unaltered by refraction. This is Malus's theorem. We have now given the equations which would determine any refracted congruence, when we are given the refracting surface. Unfortunately the equations are complicated. > 8O. The Ribaucourian congruence. We shall now con- sider some special classes of congruences. Consider the congruence formed by rays drawn from every point of a surface, parallel to the normal at the corresponding point of a surface which corresponds orthogonally to the given surface. This is the llibaucourian congruence, so called as Kibaucour was the first to consider it. We take to be the surface from which the rays are drawn parallel to the normals to the surface z. Taking the asymptotic lines on z as the, parametric lines we had f A ZQZ f QZ QZ Sl l/i*> I'"]) &2 u "2 l/ 2 /V > and Z c\, where c = ( K)~*, K being the measure of curvature on z. To bring this into accordance with our notation for con- gruences we write p. for A, and we hav 102 DEFORMATION OF A SURFACE, AND CONGRUENCES Since S^fa/i and $^ 2 // 2 // = 0, the equation which defines the developables is dudv = ; and the local points are given by w = cO, w = cO. The surface is then the central surface of the congruence, and the developables intersect it in conjugate lines with equal invariants. These lines correspond to the asymptotic lines on z, the surface which corresponds orthogonally to the central surface. 81. The Isotropic congruence. Ribaucour's theorem. We have a particular, and most interesting, case of this con- gruence, when the surface which corresponds orthogonally with { is a sphere with the origin as centre. In this case c is a constant and { corresponds orthogonally with p, itself. The congruence is z' = + w/t and is called the isotropic congruence. For the isotropic congruence, and therefore the limiting points of any ray coincide and are on the central surface. Any plane through a ray is a prin- cipal plane and any surface may be regarded as a principal surface. The lines of striction of all the ruled surfaces of the congruence lie on the central surface. In the chapter on the ruled surface [see 108] we prove that any two ruled surfaces of the congruence intersect at the same angle all along their common generator. The dev.elopables and the focal points we see are imaginary. We have proved that y = s+ ^ is a minimal surface and that p. is the unit vector parallel to the normal at the extremity of y. The perpendicular p on the tangent plane to this surface is given by p + yp = 0, * - ' that is, by p + IL ~ 0. ISOTROPIC CONGRUENCE. RIBAUCOUR's THEOREM 103 The tangent plane is therefore the plane drawn through the extremity of perpendicular to the ray of the congruence. We thus have Ribaucour's theorem that c The envelope of the plane, drawn through the extremity of the vector which traces out the central surface, perpendicular to the corre- sponding ray of an isotropic congruence, is a minimal surface '. The surface corresponding orthogonally to the sphere is therefore the pedal of a minimal surface. If two surfaces are applicable on one another, and if the distance between corresponding points is constant, we see that the line joining these points traces out an isotropic con- gruence. For if fj, is the unit vector parallel to the join of the points, and z is the vector to the middle point of the join, and 2c is tlie length of the joining line, from which equations we at once deduce the result stated. 82. W congruences. Let us now consider again the two surfaces which we denoted by z and z + gp y and consider the congruence formed by the line joining corresponding points on these surfaces. Looking at the tabular arrangement of the twelve surfaces we see that p is parallel to the normal to z at the corresponding point, and that is parallel to the normal to z + p at the corresponding point. The line joining corresponding points on the two surfaces z and z + p, being perpendicular to both p and is perpendicular to the normals to z and to z + p, and therefore touches each of these surfaces. Now if a ray of a congruence touches a surface, that surface must bo a focal surface of the congruence. For, taking z to be the vector to the surface, and p the unit vector parallel to the ray, 8^*2 = 0; and therefore, the focal points being given by S (z l + w^} (z 2 + Wfi 2 ) 14 = 0, we see that one of the focal surfaces is given by w = 0. 101 DEFORMATION OF A SURFACE, ARD CONGRUENCES It follows that z and z + p are the focal surfaces of the cpngruence we are considering. Now on these surfaces the asymptotic lines correspond. Conversely it may be shown, that if the asymptotic lines correspond on the two sheets of the focal surface the focal surfaces are z and z + gp. Congruences of this type may be called W congruences. 83. Congruence of normals to a surface. We now come to the case of congruences where the rays are normal to a surface. The theory of such congruences is of special interest in geometrical optics as well as in geometry. Instead of /z we shall write X, where A is the unit vector normal to the surface from which the rays emanate. AVe now have ^ ^ = ^\ (83.1) as a necessary condition that the congruence may be a normal one. This necessary condition is also sufficient : for if n then T-/*^ N ~/* i> ^H,^* <*V^- and we can therefore determine a function w such that n\ = /^ ; u' 2 = /*c. 2 . Let z' = z + iup., then z\yi = ZIIL + U\I& = 0, so that the rays are normal to the surface z'. The normal congruence is therefore defined by o> 21 = a> 12> (83.2) and the limiting points coincide with the focal points, and the focal planes with the principal planes. The focal planes are therefore perpendicular to one another. Conversely if the focal planes are perpendicular to one CONGRUENCE OP NORMALS TO A SURFACE 105 another the congruence is a normal one : for we see that the condition that the focal planes may bo perpendicular is and therefore, since <-i n <(^ 2 a\ a is not zero, o> 12 = o> 2l . 84. Reference to lines of curvature. We now take the parametric lines on the surface z to be the lines of curvature, and we have z l = -R f \ 1 , ^=-R"X t , t where II' and R" are the principal radii of curvature. We have co n = -R'\l a> la = a> al = 0, a> 2a = -.K"A 3, that is, o) u R'a lv a>^ = R"a^, co^ co zl u ri 0. The focal points are given by f=R', f' = R", and the two focal surfaces are now given by The equation of the developables is (R'-R")dudv = 0. As we need not consider the case where R' R" any further than we have already done we see that the equations of the developables are du = 0, tZy = 0. (84.1) For the focal surfaces we have </o' = -(R"-R')\dn + \dR'. (84 . 2) Calling this the first sheet of the focal surface, its ground form is (dR')* + (R"-R')*Mdv\ (84 . 3) and therefore the u curve is a geodesic on the ih-st sheet. Similarly we see that the v curve is a geodesic on the second sheet. 86. Tangents to a system of geodesies. Conversely if we take any surface, and draw any singly infinite system of geodesies on it, the tangents to these geodesies will generate a normal congruence. 106 DEFORMATION OF A SURFACE, AND CONGRUENCES For take a surface with tho ground form and consider the congruence formed by the tangents to tho curves v = constant, that is, by the tangents to this family of geodesies. We have /z = s l and as ~ 2 _ i ^ ^ n "i l j ^2 u > we must have c t : 12 = 0, Sn^ + 'i^iu = 0, so that C 2u ^ 0. Now c; 2 /x 1 = o. 2 and c^ = c^ 1 so that ^! = ^ 2 , and the congruence is a normal one. 86. Connexion of W congruences which are normal with W surfaces. Now let us consider the asymptotic lines on the two sheets of the focal surface. The vector to the first sheet is and we have o' 1 =-R' 1 X, c' 2 = (II" R') and therefore (R r - R") X 12 = R'\ A 2 - R' The equation of the asymptotic lines is if V is the unit vector parallel to the normal at the extremity of z'. Now A x is parallel to V, and therefore the equation of the asymptotic lines is dz'd\ = ; that is, S((R"-R') \ 2 dv-\dR') (\ u du + X l2 dv) = 0. We have, since AjA 2 is zero, A n A 2 = A T X 12 = ^ A 'f -r (R f R")> XX n = X^, XX ]2 = 0, W NORMAL CONGRUENCES AND W SURFACES 107 and therefore the equation of the asymptotic lines on the first sheet is X*R' l d<u,*-\lR" l <W = 0. (8G . 1). Similarly we see that the asymptotic lines on the second sheet of the focal surface are given by \lR' 2 du 2 -\lR" 2 <lv 2 = 0. (86.2) The necessary and sufficient condition that the asj mptotic lines on the two sheets may correspond is therefore that R f and R" may be functionally connected. We thus have the theorem that in a TF congruence, if it is also a normal one, the surfaces which intersect the rays orthogonally have their radii of curvature functionally con- nected : that is, they are W surfaces. 87. Surfaces applicable to surfaces of revolution, and W normal congruences. We saw ( 84) that the ground form of the first sheet of the focal surface of a normal congruence was (dR'Y + (R"-R')*an<lv\ (87 . 1) and similarly we see that the ground form for the second sheet is (dR")* + (R"-RJa n du 2 . (87 . 2) If the congruence is also a W congruence we know that the ground forms of the first and second sheet are then respectively (<fcfl')2 + ^ (R'jfdtf, (87.3) (dR")* + (<t>'(R')Y*dtf. (87.4) The two sheets are therefore applicable on surfaces of revolution, the u curves on the first sheet corresponding to the meridians, and the v curves on the second sheet. Conversely, if we have any surface applicable on a surface of revolution, the curves which correspond to the meridians will be geodesies, and the tangents to these curves will therefore trace out a normal congruence which will be a W congruence ; and the surfaces which cufc the rays orthogonally will bo W surfaces. 108 DEFOKMATION OF A SURFACE, AND CONGRUENCES If the surface is one of constant curvature we need to solve an equation of Riccati's form to obtain the curves which correspond to the meridians, but in other cases we can find the curves by quadrature. An interesting property of any given W surface, which is not of constant, curvature, is that we can find the lines of curvature on it by quadrature. For we can find the two sheets of the focal surface, and on these sheets we can find by quadrature the curves which correspond to the meridians. These curves will have as their correspondents on the given W surface the lines of curvature. This theorem was discovered by Lie. 88. Surfaces of constant negative curvature. Returning to the ground forms of the two sheets of the focal surface we see by aid of the formula whim the ground form is (hr + ffidr 2 , that, K' denoting the measure of curvatuie on the first shoet, Similarly we find for the measure of curvature K" of the second sheet K" + <^> </') J 4 -f- [(0 (R'))'' <//' (R')] ^ (88 . 2) since K' = /('), R'-f(R' } = |^ If the two sheets are applicable on one another at correspond- ing points we must have K' = K" and therefore we must have 0" (IV) (j> (K) - (f (R'))*. (88 . 3) Taking the upper sign wo see that where a and b are constants. SURFACES OF CONSTANT NEGATIVE CURVATURE 109 We now see that R"-R' -a, (88 . 4) from the equations It' -f (R f ) = R" =f(R'). The measure of curvature is found to be a"" 2 from the formula 6" (R'\ A " + ?wr - (S8 - 6) The two sheets have then the same constant negative measure of curvature a~ 2 , and the distance between the corresponding points is equal to the constant a. We therefore see how, when we are given a surface of constant negative curvature, we can construct another surface of the same constant curvature. We find a system of geodesies on the given surface this involves the solution of an equation of Riccati's form- and draw the tangents and take a constant distance a along the tangent: the locus of the point so obtained will be the surface required. CHAPTER VI CURVES IN EUCLIDEAN SPACE AND ON A SURFACE. MOVING AXES 89. Serret's formulae. Rotation functions. Let A, p, v be three unit vectors drawn through the origin, respectively B parallel to - the tangent, principal normal and binormal of a curve. We see from the figure that d\ = /zc/6, dv = /jidrj, where de and drj are the angles between neighbouring positions of the tangents and osculating planes respectively in the sense of the figure. We thus* have where the dot denotes differentiation with respect to the arc of the curve, and p and a- are the radii of curvature and torsion respectively. We thus have P - 1 t a SERRETS FORMULAE. ROTATION FUNCTIONS 111 and therefore, since X/JL = 0, p.v = 0, we have /iX = - , jw = --- , ftp = 0. - ' p - o* < ' It follows that v^-P- (89.1) p These are the formulae of Serrct. If we were to take unit vectors through the origin mutually at right angles, the first, X, parallel to the tangent to the curve, and the second, /z, making an angle (/> with the principal normal, wo could easily deduce that X = /jLT vq, /i = i>^ X/', v=- Xq fJLpy . ; 1 sin cos where p </> + , q = -- r , r - - o- p p More generally, if X, /^, v are three unit vectors mutually at right angles which are given angular displacements pds, qds t rds, we have \ p.r-~vq, fi = vl> \r, v Xq p.^), (89.2) as we see from the figure. The functions p y q, r may be called rotation functions. If cods denotes the angular displacement which the vectors regarded as a rigid system receive, where we can write our equations in the more elegant form X = coX, (JL = a)//, v = <>v. (89 . 3) 90. Codazzi's equations. It will be useful to consider a more general displacement. Let the vectors X, /*, v regarded as a rigid system receive three angular displacements co'du, ("dv, co'"dw. We then have X 1 = o/X, X 2 = oFx, X 3 = a/^X ; and therefore co'X^, + o?^X = ^'X^^ 4- a)'\X ; 112 CURVES IN EUCLIDEAN SPACE or Wo/ 7 *- Fo>"o7x = V(<o'\ -cog X, that is, Fo/^/'X = F(a) // 1 -a)' 2 ) X. (90 . 1) Wo have exactly the same equation for p. and therefore wo have identically ^// = ^ _ ^ Similarly we obtain two other vectorial equations, and we have o)'" 2 o>" 3 = o>" o>'", o>' 3 co'"! = a/"**/, a/'j a/2 = a)'a>". (90.2) Suppose now that the vectors X, p,, v instead of being drawn through the origin are drawn at the extremity of the vector z } which depends on the three parameters u, v, w. If we regard the extremity of the vector z as the new origin then we may say that the linear displacements of the origin are z^lu^ z< 2 dv y z.^dw. Let -'X + We therefore have so that Similarly we obtain two other sets pf equations : CODAZZiS EQUATIONS 113 If we ignore the parameter w, we have the six equations : C*-C\=l>'*l"-P"l' + l"t'-<l't"' (90.3) These are the equations of Codazzi of which Darboux makes so much use in his Theory of Surfaces. 91. Expressions for curvature and torsion. Returning now to the case of a curve, Serret's equations may be written where A = coA, fi COJJL, v cov ; >- A " 00 "~ a- p If is the vector which describes the curve to which we arc applying Serret's equations we may write (91.1) where i ', t", i" are three fixed orthogonal vectors through the origin, so that z' , z" , z'" are the Cartesian coordinates of any point on the curve. We have z ~ A and therefore or . a pz y v <rpz + <rpz + 8. P Denoting the components of the vectors A 3 /z, v with respect to i \ i"> if" in the usual way, wo know that " '" A', A", A and therefore and v , v z', z", z" z', z", I'' z', s", z' (91.2) 1 p* (91.8) These are the usual formulae in the theory of curves. 2813 Q 114 CURVES IN EUCLIDEAN SPACE If we take, as is more usual, x, y, z to be the Cartesian coordinates of any point on the curve and regard them as functions, not of the arc, but of any variable, we see that , y, 1 * "2 // <j / / i/'j -U , * /y, * (91 .4) (91.5) 4 , & * , y, ^ 92. Determination of a curve from Sorret's equations. We must now show how the equations .__/* ._ X v ___/* p ' p cr y <r determine the curve when we are given the natural equations of the curve ; that is, when we are given p and <r in terms of the arc. Any unit vector may be written sin cos $ . X + sin 6 sin (f> . p. f cos . v. Expressing a fixed vector in this way, and noticing that there can be no relation between the vectors A, p,, v of the form pX + qp + rv = 0, where p, q, r are scalars, we find, by aid of Serret's equations, DETERMINATION OF A CURVE FROM SERRET's EQUATIONS 115 Let ^ = cot I e"/', u then we find that \/r = L (\// 2 I) ^- (92.2) This is an equation of Riccati's form. When we have solved it, we know & and 0, and thus the position of a fixed vector with reference to A, /z, v. When we have thus found three fixed vectors, with reference to A, /z, v, we know A, /z, v in terms of the arc. When we have obtained A in .terms of the arc we can find z by aid of the equation \ t (92.3) It must now be shown how, when we are given any curve in space, any other curve, with the same natural equations, can by a mere movement in space be brought into coincidence with the given curve. IF A , /z , */ denote the positions of the vectors A, /z, v when the arc s is equal to s or, say, to zero, then we see, by repeated applications of Serret's equations, that ^c'Ao + e'Vo+c"'^, (92.4) whero the coefficients of A , /z , */ are known series in powers of 8. By a mere rotation wo can bring A , /z , z/ into coincidence with the tangent, principal normal, and binormal at the point from which we measure the arc on the given curve. It follows that A, /z, v will be unit vectors coinciding with the directions of the tangent, principal normal and binormal at the point s on the given curve. A mere translation will therefore bring tne cfirve into coincidence with the given curve when the required rotation has been carried out, since we have z - A, z 9 = A, > * and thus c'~0 + a where a is a fixed vector, that is, a Vector not depending on the arc. 116 CURVES IN EUCLIDEAN SPACE 03. Associated Bertrand curves. The right helicoid. .Let us now consider the curve defined by z' = z + kii. (93.1) where k is some function of the arc, and let us find the conditions that the two curves defined by z and z* may have the same principal normal. We have y = (x+pk + k- X\\ ds and therefore A' = (\ + pk + k(^- X A 6#S p')dj Since XV = 0, we must have k equal to zero [i.e. k a constant]. Again, differentiating with respect to the arc ', and therefore /cA( TV ) + ( 1 -ITT? > *\db / \ /ds* - p 2 ) - T- 2 a- ds 2 ( I Q ^/ ^ ^ Eliminating , and y-^ wO obtain p A; p or /; &' and integrating we have " -f = 1, (93 . 2) where k' is a constant introduced on integration. A curve satisfying the above equation is called a Bertrand curve. We see that the property of a Bertrand curve is to be associated with another Bertrand curve having the same principal normal, the distance between corresponding points being the constant k. If a Bertrand curve has more than one corresponding curve ASSOCIATED BERTRAND CURVES. THE RIGHT HELICOID 117 it will have an infinite number of such curves and will clearly be a circular helix, for p and cr will each be constant. We can immediately deduce that the only ruled minimal surface is the right helicoid. For consider the curved asymp- totic line on a ruled surface. We know that the osculating plane of any asymptotic line on any surface is a tangent plane to the surface. The generator of the ruled surface therefore lies in the osculating plane of the other asymptotic line through any point on it. If the surface is a minimal one it must therefore be a principal normal, and since an infinite number of asymptotic lines cut any generator orthogonally the asymptotic lines must be circular helices. The surface is therefore a right helicoid. 94. A curve on a surface in relation to that surface. We now pass on to consider the curves which lie on a given surface. Since such curves are defined by a relation between the parameters u and v, and since z y the vector of the given surface, is a function of these parameters, we are really given 2 in terms of one parameter along the curve defined by an equation F (u> v) = 0. But since we want to consider the curves in relation to the surface we proceed by a different method. We have the formulae where X is a unit vector parallel to the tangent to the curve, fji a unit vector parallel to the normal to the surface and making an angle with the principal normal to the curve ; and we have seen ( 89) that sn p where p and a- are the radii of curvature and torsion of the curve. We know that 118 CUEVES IN EUCLIDEAN SPACE we can therefore easily verify the formulae and from these formulae we deduce ///^ 1 1 = (e/2 22 It follows that and that A/cs = h (94.2) (94 3) But and and therefore cos 6 (JL\ r L p fjLZ ltfi\ = /2 n -i6 2 -f 2f2 12 uv + f! 2 (94.4) (94 . 5) Wo have thus expressed the two angular velocity com- ponents p and r of the curve under consideration in terms of the derivatives of the parameters u and v with respect to the arc and the functions e, /, g and fl n > /2 12 , /2 22 . We must consider the remaining component q. As the vectors X, //, v are displaced from their positions at P to their positions at P', a neighbouring point of the curve under consideration, we may consider that they are displaced along the geodesic PT and then along the geodesic TP 9 . As we pass along Py the displacement qds is zero and as we pass along TP' the displacement qds is also zero. The CURVE ON SURFACE IN RELATION TO THAT SURFACE 119 total displacement qds is therefore just the angle P'TM: that is [39] i g= (94/6) Pff since the geodesic curvature of the curve is defined by the formula Pff = Lt, P'TM We should notice that unlike p and r the angular velocity q depends on the first ground form only and the derivatives of u and v and not on 4f2 n , /2 r2 , /2 22 . We have proved earlier (36 . 3) that We express this formula in a more convenient form for some purposes without the aid of the differential parameters by P, {,t 11 Fl-2a llj F i ^ + a M ^} where F(v,v) is the equation of the curve, or, since ^16 + 1^= and F } (u+{ll\} u*+2 {121}^+ [221] v 1 } + F tji (i)+ {112}u 2 +2 {212} uv+ {222} v*) + F. n u* + 2F.uUV + F. 22 v* = 0, and a n in the form (94 . 8) = h Pa V, (94 . 9) 120 CUKVES IN EUCLIDEAN SPACE Wo have thus found expressions for the angular velocities ; 1 sin cos0 .^ A x ,, = # + -, V = -f. 'r=-f> () along the curve in terms of the derivatives of u and v and the functions which define the ground forms. We notice that p and r depend only on the first derivatives, but q depends on the second derivatives and is the geodesic curvature of the curve. We have seen [ 49] that the curvature of the normal section of the surface in the direction of the tangent to the curve is given by We thus have Meunier's theorem that * = i- (*. 10) p It The expression -}- % (94.11) is the same for all curves having the same tangent at the point under consideration. It is therefore the torsion of the geodesic curve which touches the curve at that point. 95. Formulae for geodesic torsion and curvature. We can find another formula to express the torsion of the geodesic by aid of the formula already proved & + (R 9 + R") *p. + H'H"jP = o. Since z = X and (i = i/p Ar, we have 1 - (R f 4- R") r + R' R" (p* + r 2 ) = 0, that is, p* + (F ~ r ) (JET ~ r ) = * (95 !) If we take the parametric lines as the lines of curvature, so that _ cos 2 Q sin 2 T ~~W~+ R" ' this becomes p = cos sin d (-^ ^777 j or -h - = cos 0sin0(-g> jp)' (95.2) FORMULAE FOR GEODESIC TORSION AND CURVATURE 121 Since q is the angular velocity about the normal to the surface, as we pass along the curve we are considering, we see that < t where q f it, + q" v is the angular velocity about the normal of the rigid system made up of the normal and the tangents to the two lines of curvature. We thus have the formula for the geodesic curvature l - = - 6 + (/ (L + q"v. (95 . 3) P<J We have and therefore r Zpq (95 . 4) depends only on the first derivatives of the parameters u and v, and so is the same for all curves on the surface having the same tangent at the point under consideration. This theorem is due to Laguerre. In connexion with the formulae 1 sin tan where R is the radius of curvature of the normal section of the surface in the direction of the tangent to the curve, it is useful to remember that if a particle describes a curve on any surface with velocity V, the acceleration normal to the path 72 and tangential to the surface is -- Pff 96. Surfaces whose lines of curvature are plane curves. So far the curve we have been considering has been any curve on the surface: suppose now . that it is a line of curvature. 122 CURYSS IN EUCLIDEAN SPACE 'X, We 6a*ve p = 0, as we see from the formula ' and therefore + - ^ 0. (96 . 1 j 0" If therefore the line of curvature is a plane Qurve its plane makes a constant angle with the surface all along it ; and conversely if the osculating plane at each point of a line of curvature makes the same angle with the surface the line of curvature is a plane curve. We now propose to find the form of a surface if all its lines of curvature are plane curves. Let a be a vector perpendicular to the plane line of curva- ture along which only v varies so that a depends on u only. Similarly let ft be a vector perpendicular to the plane line of curvature along which only u varies. In accordance with our general notation in the theory of surfaces, let A be a unit vector normal to the surface at the extremity of the vector z. We have, since the parametric lines are lines of curvature, and as as. 2 0, /J^ = 0, we also have aA 2 = 0, /?A 1 = 0. It follows that oc=pX l -\-qX, /8 = rA 2 -fsA, where p, q, r, s are scalars. We thus obtain the two equations l = 0. (90 . 2) Now since SX U X 1 X 2 = 0, as the lines of curvature are conjugate lines, ? 2 = 8 i = 5 and as there can be no relation between Aj and A 2 of the form g = 0, SURFACES WITH ONLY PLANE LINES ftF CURVATURE 123 where a and b are scales, we must hav$A B = i, * = .'.' (9G.-3) p r r p 8(7 It follows that (log>) 12 = (logr) 12 = --' , and therefore we may take p = F(u)e', r=f(v)e, q = F(u)J0 l9 8=f(v)e'0 2 . (00. 4) We also have 12 -f O l 2 = 0, so that \ 2 +02 A i + 0i X 2 = ' (96*5) and 0^ + 0^ = 0. (96.6) Let us now start again with these two equations. We see that 0i 2 + M 2 = tells us that e = f (u) + </> (v) ; and, since \ } A 2 = 0, the lines of curvature being at right angles, the equation tells us that (A?) + 2 2 A = 0, so that A!+6- 2tf jP(w) = 0, A^+e~ 2 ^0(v) = 0. (96.7) We can now so choose the parameters that Xf+- 8 =0, A+- M = 0. The spherical image of the surface is therefore given by where A~ l U being a function of u only and F a function of v only. 124 CURVES IN EUCLIDEAN SPACE But, from the expression for the measure of curvature of the surface c fa* = A 2 du 2 + B*dv An * /I ^7^\ * /I *A\ KAB + --(-_) + -- ( - - ) = 0. <)u\A <>u/ 3v\B <>v/ We must therefore have and therefore 1 = UU-U*+ VV- F 2 + UV+ VU. (96.8) It now easily follows that without loss of generality we may take fj C0 sec a cosh u, V = cot a cos v, e cosh u- cos a cosy so that c = - : -- (96 . 9) sin a v / If p is the perpendicular on the tangent plane to the surface of which we have found the spherical image we have 2) + Xz = 0. It follows that since A^ = 0, Xz 2 = 0, A r c 2 = 0, A 2 x? 1 = ; and therefore ^) 12 -I- fl^^ 4- fl^ = 0, that is, Bince d 12 + ^ t 2 - 0, pe e =U+V, (96.10) where [7 is a function of u only and V a function of v only. We know that , fi where e = = 0, A 1 A 2 =.0, A2+e- 2 *= 0, cosh u cos a cos v sin a and therefore we can find A by the solution of equations of Riccati's form. SURFACES WITH ONLY PLANE LINES OF CURVATURE 125 Wo see that (cosh u cos ex cos v) A = sin a sinh ui + sin a sin vj + (cos a cosh u cos v) k. where i,j, k are iixed unit vectors at right angles, will satisfy the conditions ; and we know that any other possible value of the vector A can be obtained from this vector by a mere fixed rotation. The surface may therefore be regarded as the envelope of the plane x sin ex sinh u -f y sin a sin v 4- z (cos a cosh u cos v)'= U+ V. (96. 11) 97. Enneper's theorem Let us now consider a curve which is an asymptotic line on the surface. We have y ? = for an asymptotic line and therefore cos ft P If p is infinite the asymptotic line is straight and therefore the surface is ruled. Leaving aside the case of ruled surfaces, cos is zero and therefore = , that is, the osculating plane of an asymptotic Lt line is a tangent plane to the surface. For an asymptotic line the angular velocities are 1 1 ^ r=0, p* + (-j^ r\ (-^,, r\ = and the formula gives I 2+ _J_ 7> = 0) (97.1) that is, the torsion is \/ K. This is Enneper's theorem. We also see that the geodesic curvature of an asymptotic line is just the ordinary curvature. 98. The method of moving axes. If we now return to the equations of Codazzi (90 . 3), which are the foundation of a considerable portion of Darboux's method of treating problems 126 CURVES IN EUCLIDEAN SPACE of differential geometry, a method which is in effect the method of moving axes, we may take ', " to be zero. The rotations are p', q', r' ; p" t q", r", and the translations '' 'n't Q > "> V? y an d the connexions are q"r' t <l',-<l'\ = r'p"-r"p', 7's-V't - r'S"-r"t', !>' l" ~2>" l' - q't"-q"t'. The displacements of a point whose coordinates with reference to the moving axes are x, y, z are, with reference to fixed axes with which the moving axes instantaneously coincide, dx + gdu + "dv y (r'du + r"dv) + z (q'du + q"dv), dy + rj'du + rj"dv z (p'du+p"dv) + x (r'du + r"dv), dz x(q'du + q"dv) -f y (p'dii +p"dv). Thus for a curve on the surface making -an angle o> with the axis of x ds cos co = 'du+ "dv, dsama> = i]' du + r)" dv. (98 . 1) A point on the normal to the surface and at unit distance from the surface traces out what we call the spherical imago of the surface. Thus the spherical image of the curve is given by da cos = q'du + q"dv, dcr sin 6 = p'dup"dv. (98 . 2) The direction of the line element conjugate to the line element whose direction is co is 6 -f - , and therefore the two i elements du, dv and Su, Sv will be conjugate if ?8v, + t"8v_T,'tu + j'8v ( . 2/du+p"dv ~ q'du + q"dv V ' ; The asymptotic lines, being the lines traced out by self- conjugate elements, will therefore be given by dv ' ( ' p'du + p"dv THE METHOD OF MOVING AXES 127 The spherical image of the surface will be given by d(T 2 = (p'du + 1>" dv) z + (q'du + q"dv)*. (98 . 5) The principal radii of curvature and the lines of curvature will be deduced from the fact that the point whose coordinates are R will have no displacement in space and therefore (' + Rq') du + d" + R<f) dv = 0, It follows that the measure of curvature will be given by rt/u"__/V r/ r " Here we should notice that the translation functions depend only on the ground form, as e = (I') 2 + (I")', / = IV + !' V. .9 = (V) 2 + (O 2 , and that r' and ?" can be expressed in terms of the translation functions, so that we see again and very simply that the measure of curvature is an invariant. If the surface is referred to parametric lines at right angles we may take ( ; 6 2 __ A 2 du 2 + JPdv f2 9 and ' = A, " = 0, rj' = 0, r?" = B. We then have r' = ~ , r" = ~ , ti A and at once deduce the formula If we refer the surface to the lines of curvature as para- metric lines we have p r = q" = 0, and the principal radii of curvature are A R R'=-^ t , R" = ^' (98.8) 99. Orthogonal surfaces. To illustrate the employment of moving axes depending on three parameters we might consider the case of orthogonal surfaces t u = constant, v = constant, w = constant, 128 CURVES IN EUCLIDEAN SPACE and take as axes the normals to these three surfaces at a point of intersection. We have " = "' = 0, i/" = rf = 0, { = f" = ; and we may write g ^" 7^ '" . The equations satisfied by the translation functions now become ^ = _/,, & = ^/, V; , = _^/', , 1 = r ", fi = -^/'", ^ = 9X", ^,/' + ,r'" = 0, f /" + ft/ = 0, 9i / + ^/" = 0. We therefore have ^' = 0, a " = 0, r'" = 0, and we have the well-known theorem that the lines of curvature on the surfaces are the lines where the orthogonal surfaces intersect them. We shall return to the theory of orthogonal surfaces later and so shall not pursue the study further here. CHAPTER VII THE RULED SURFACE 100. Let a vector trace out any curve in space, and let X', yn', i/ f be unit vectors drawn through its extremity, parallel respectively to the tangent, principal normal and binomial of the curve. Let X == cos 6 .X' sin sin . p! -h sin cos </> . v \ p, = cos . fj. f + sin . i/', p = -sin 0.X' cos sin . /*' + cos cos 0./, (100. 1) then X, //, *> will also be unit vectors mutually at right angles. On the unit sphere, whose centre is the origin, vectors parallel to these two sets will cut out the vertices A, B, C and X, F, Z of two spherical triangles as in the figure. Let - and - denote respectively the curvature and torsion P o- of the curve and let 1 , sin d> , cos , , rt . ? / = 0+ , 9'= , r' = , (100.2) <r ^) /> and ^) = p' cos 5 + r' sin ^, (/ = q' 0, r = r' cos fl p' sin 0. (100.3) 2843 S 130 THE RULED SURFACE By aid of Serret's equations we see that \ = TfjL qv ) (Lpv r\, v = q\pn, (100.4) so that p> </, r are the rotation functions for the moving triangle XYZ. The dot above any symbol denotes that it is the symbol differentiated with respect to the arc of the curve traced out by : we denote the arc by v. 101. The ground form and fundamental magnitudes.* Let z = +uA, so that u is the distance of the extremity of the vector z from the extremity of the vector As u and v vary, the vector z will trace out a ruled surface of the most general kind if Q and are functions of v. The curve traced out by will lie on the ruled surface : it is called the directrix of the ruled surface. Any curve on the surface may be taken as directrix. We have Z 1 = A, z. 2 = A' + u A cos \ + urp (sin + uq) v. The ground form of the ruled surface will be ds 2 = du* + 2 cos Qdudv + (u z (q* + r 2 ) + 2 u? sin 0+1) <t/ 2 . (101. 1) We may write ? = ]\f cos ^ r = j|/ s j u ^ ? when the ground form becomes efe 2 = du z + 2 cos 0rftt dv + (u 2 -J/ 4 + 2 u3/ sin cos ^ + 1 ) c/t' 2 , (101 .2) so that q and r are given when the ground form is given. The function h is given by k 2 = u 2 JI/ 2 + 2uJlf sin cos ^ + sin 2 0, = (uJf + sin cos ^/r) 2 + sin- sin 2 \/r. (101.3) The angle between two neighbouring generators is Mdv, (101.4) and the shortest distance between them is sin 0sin ^ dv. (101 . 5) The unit vector whose direction is the shortest distance is cos i/f p, + sin ^ v. (101 . 6) * [Soo also 22.] THE GROUND FOKM AND FUNDAMENTAL MAGNITUDES 131 Since z l = A, # 2 = cos 0X + uM sin-v/r/j (uM cost/r + sin d) j>, we have r ? 2 = (uM cos \/r + sin. 6) JJL -f uJf sin ^/r j/. The unit vector normal to the surface at the extremity of z is therefore Z, where Z = /6' 1 [(uJfcos>/r + sin5)/z + ^^/sin\/rj/]. (101 . 7) If we calculate 3 n , 12 , s 22 we deduce, by aid of the formulae J2 11 = z^Zj J2 12 = Z^L, J2 22 z^Z y that /2 n = 0, /2 12 = /fW sin ^sin 0, /2 22 = p/^ 4- /^ -1 ((it 2 J/ 2 + u3f cos *//- sin 6) r/r + uM sin *// sin + J/cos(9sin^(sin^-ufl)). (101 .8) We may write N for M cos \/r sin when we are only con- sidering the ground form. 102. Bonnet's theorem on applicable ruled surfaces. We saw that one of the most difficult problems in the Theory of Surfaces was, given the ground form, to determine the surfaces in space to which the form was applicable ; and wo saw that the solution of the problem depended on a partial differential equation of the second order. In general we can- not solve this equation, but there is a striking exception in the case of the ruled surface. Let us first consider a theorem on ruled surfaces. If on the surface with the ground form a^dx^dx^ the curves x 2 = constant are geodesies, we must have {112} = 0. If the curves x 2 = constant are asymptotic lines we must have /2 11 = 0. If both these conditions are fulfilled the surface is ruled; that is, if n n = and {112} = 0, . (102.1) the surface is ruled and the generators are <r 2 = constant. (102.2) Now suppose that we have a second ruled surface with the same ground form and therefore applicable on the first surface, and suppose if possible that its generators are not the lines X) = constant. 132 THE RULED SURFACE We can therefore choose our coordinates so that the two surfaces will have the same ground form and that in the first surface oc 2 = constant will be the equation of the generators and on the second surface x 1 = constant will be the equation of the generators. We have /2 n = and /2' 23 = 0, and as for the two surfaces fl u fl^ n^ and /2' n /2' 22 -/r^, are the same, we must have /2* 2 =/2'?.>. (102.3) From Codazzi's equation (48 . 4) for the two surfaces we have It is therefore possible to satisfy Codazzi's equation for the given ground form with {112} = 0, {221} = 0, (102.4) by taking fl u and /2 22 both zero : that is, it is possible to find a surface with both systems of asymptotic lines straight lines; that is, to find a quadric applicable to the given ground form. Unless then the form du* + 2 cos Odu dv+(M 2 u* + 2Nu+l) civ* (102 . 5) is applicable to a quadric, the generators of any ruled surface which is applicable to it must be v = constant. (102 . 6) This is Bonnet's Theorem and Bianchi's proof of it. When therefore the ground form is given in the form d** = du* + 2 cos 0dudv+(M 2 u 2 + 2Nu+ 1) dv 2 , (102 . 5) we know that, leaving aside the case of quadrics, the surfaces which are ruled and applicable. on it must be generated in the method we have described [so that their rectilinear generators are applied to its rectilinear generators]. BONNETS THEOREM ON APPLICABLE RULED SURFACES 133 When the ground form is given we are given q and r. Wo may take p as any arbitrary function of v. We then know p and <r of the directrix, and so can find it by the solution of Riccati's equation. Similarly we obtain X and thus find the ruled surface. 103. Ground forms applicable on a ruled surface. If we are given the ground form of a surface, how are we to decide whether it is applicable on a ruled surface? It will be applicable on such a surface if the ground form can be brouht to the form du* + 2 cos Odudv+ (M 2 u* + 2iVW 1) <lv 2 , (102 . 5) where $, J/, and N are functions of v only, but unless these are given functions of the parameter the general method will not immediately apply. This is the question we now wish to consider. The expressions du , dv r and -T-, as as where u and v are the parameters of a point on the surface, are tensor components. We may denote them by T l and T l . The difficulty of the tensor notation feomes in when we want to express the power of a tensor component with an upper integer. Thus the square of T 2 would have to be written T 2 T 2 , and in calculations this is inconvenient. We therefore generally write the above two components as and ?/ and try just to remember that they are tensor com- ponents when we apply the methods of the tensor calculus/ The equations of a geodesic are ( 38) + {112} f + 2{122J^+{222], / 2 = 0, and <l-. f *t +n li its C 5a; 1 'do--/ ^ = *li + ) ,te*. 134 THE RULED SURFACE The equations of a geodesic may therefore be written + 1121} +[221] ?=(), (103.1) Now these equations are very simply expressed in the tensor notation by T 1 T.\ + r^ = 0, T^T:\ + y-T.^ = 0. (103 . 2) The equation of the asymptotic lines is given by n ll T i x i + 2n u T l T*+n M 'i*T* = o. (103.3) Now remembering that on a ruled surface one of the asymptotic lines is a geodesic, and taking the tensor derivatives of this equation, we have T* (/2 la 2V, + 12, 2 T.\ ) = 0, (103.4) and /2 ll . 2 7 T1 2 Tl + 2/2 12 . 2 T 1 ? T2 + /2 2 ,. 2 7 T2 r + 2^(/2 11 TJ a +/2 12 ^ 2 ( + 2r^/2 12 IV 2 + /2 22 ^ 2 ) = 0. (103.5) Multiplying the first equation by T l and the second by T 2 , and adding, and making use of the equations for a geodesic, we see that if the surface is ruled we have for the equations of that asymptote which is a generator <>, (103.6) . 2 V i = 0. (103.7) If we write these two equations (a, b, c, dlg, rj) ?> = 0, the eliminant is (Salmon, Uiylier Algebra, 198) a*C J - 6 abBC* + G acC (2 B*-AC) + ad (6 ABO- 8 + 9 b i2 ACr z - 18 6c JJ3C+ 66d4 (2 B*-AC) + 9 c -fycdBAt + dtA*^ 0. (103.8) This vanishes for a ruled surface. GROUND FORMS APPLICABLE ON A RULED SURFACE 135 Now we know that n n a,,-ni 2 =K(a n a^-ai 2 ) t and, since an arbitrary function is needed to express /2 n , /2 22 , fl vj , in terms of the parameters, there can only be one other eq nation connecting these functions. Applying tensor derivation to this equation we have, using Codazzi's equations, (103.9) Wo thus have three equations, viz. these two and the oliminant we have found. We conclude that this system must be complete if the surface is ruled. For if another equation of the first order in the derivatives of /2 U and fi^ could be obtained the function /2 12 is known in terms of /2 n and /2 22 wo could obtain /2 n and /2 22 by quadratures, and no arbitrary function woulTl appear. This method, though tedious actually to carry out, will enable us to determine whether any given ground form is applicable to a ruled surface. 104. Case of applicability to a quadric. We must now consider the ground form du 2 + 2 cos dudv -f (M 2 i<? + 2Nu + 1) cZi; 2 , as regards its special form when it is applicable to a quadric. The Cartesian coordinates of any point on a fixed generator of a quadric may be taken to be av + b ' J av + b ' av + b A where the variable v denotes distance on that generator. We have similar expressions for the coordinates on any other generator ; and the variables v and v' of the points which lie on the same generator of the opposite system will be connected by a bilinear equation. It follows that if P l is a point on the first generator and P 2 136 THE KULED SURFACE its correspondent on the second generator the direction cosines of their join may be taken as 1) D D ' where a^ , b lt c l , <l } , a, 2 , 6 9 , 2 , d t> are linear functions of v and jD 2 = dj (af 4 &1 +c|) +d| (a | -f 6J + c| ) -2c/ 1 rZ 2 (r/ 1 2 + 6 1 6 2 -h^ 1 ^). (104 3) The coordinates of any point on the quadric may then be expressed in terms of u and v in the form 61 (? 6.j ^i - 1 -" (104.4) It follows that J/ 2 , iV/>, coaOD are rational functions of ?; which can be calculated, and that J) 2 is a quartic in v. 105. Special ground forms. Binormals to a curve. Line of striction. We have found in 100, 101 the chief formulae required in the study of the general ruled surface. When the ground form is given we are given q and r, and we find the different ruled surfaces which are applicable to the form by varying p. This generally means that we vary 0, the angle of inclination of the osculating plane of the directrix to the corresponding normal section of the surface. We cannot however take (/> to be zero unless the ground form is special : for, if is zero, q -f- 6 is zero : that is, M cos \ls + = 0; which would give the special ground form cfc 2 =i <lu* + 2 cos Odudv + (M 2 u 2 - 2 6 sin 0u -f 1 ) dv*. (105.1) Thus the binomials to a curve in space trace out a ruled surface with the special ground form d** = du* -f ~ + lrf v 2, (105 . 2) where cr is the radius of Wsion. THE LINE OF STRICTION , 137 If we take as directrix an orthogonal trajectory of the generators, is , and the ground form is 2 d8 z = cZu 2 + (4/ 2 u 2 + 2 Mu cos ^ + 1 ) cfo 2 . * (105.3) In seeking the surfaces which are ruled and applicable to this form we may take for one of them cf> = i . The directrix 2 of this surface will be an asymptotic line and the surface will be generated by the principal normals of this directrix. We obtain the equation of the line of striction from the ground form itself. We have to find for given values of v and dv the values of u and dw which will make du? + 2 cos Odudv + (JfcPu 2 + 2 M n cos ty sin 6 4- 1) dv 2 least. Clearly we must have dtb + cosOdv 0, 3/K, -f cos \//- sin = 0. The equation of the line of strictiou is therefore J/u + cos^sinfl = 0. (105 .4) Let us take the line of striction as the directrix. We must then have cos i/r sin = 0. (105.5) We cannot have sin 6 equal to zero unless the shortest distance between neighbouring generators vanishes : that is, unless the ruled surface is a developable. We must therefore have in general, when the line of striction is taken as the directrix, ^ = - , and therefore q = 0. 2 It follows that 6 = , i.e. that $, the rate of increase of P the angle at which the line of striction crosses the generators, is equal to the geodesic curvature of the line of striction. It follows that the line of striction will cross the generators at a constant angle if, and only if, it is a geodesic. In this case the ground form will be tte 2 = du 2 + 2 cos 0(dudv 4- (Jf ?u 2 -f 1) dv*, (105.6) where a is the constant angle of crossing. 138 THE RULED SURFACE 7T If a = - the form will be applicable on the surface gener- ated by the binormals of a curve in space. 106. Constancy of anharmonic ratios. Applicable ruled surfaces and surfaces of Bevolution. We shall now con- sider the equation of the asymptotic line which is not a generator. The equation is 2f2 n du + f2 22 dv = 0. (106 . 1) Referring to the values given for /2 12 and /2 22 wo see that this is an equation of Riccati's form. It follows that the equation of an asymptotic line is <"> where a, /?, y, S are some functions of v only, and k is an arbitrary constant. We thus see that every generator is cut in a constant anharmonic ratio by any four fixed asymptotic lines. We also notice from the property of Riccati's equation that if we are given any one asymptotic line we can find the others by quadrature. We have also seen in 101 that the normal to the ruled surface is parallel to uH(cos tyfj, + sin tyv) + sin dp. It follows that the anharmonic ratio of four tangent planes through any generator is (Ut-uJXut-uJ - 9 v ' that is, the anharmonic ratio of the planes is the same as that of the points of contact. Suppose now that P is any point on a generator, and that the tangent plane at P intersects a neighbouring generator in P'. Then in the limit PP' is the element of the asymptotic line at P. It follows that the asymptotic lines through four points on a generator intersect a neighbouring generator in ANIIARMONIC PROPERTIES 139 the cross ratio of the tangent planes : that is, in the cross ratio of the points of contact. We thus have a second and more geometrical proof of the theorem that every generator is cut in a constant cross ratio by four fixed asymptotic lines. This theorem also is duo to Bonnet. The condition that the normals to a ruled surface, at two points u l9 u 2 on the same generator, may be perpendicular is u^u^M** (M! + w- 2 ) Msin <9cos i/r + sin 2 = 0. (106 . 4) The points are therefore corresponding points in an in- volution range whose centre is on the line of striction. No ruled surface exists which is also a surface of revolution except the quadric of revolution. We see this at once by considering a surface of revolution in relation to any meridian line. The asymptotic lines, through any point on this line, must be symmetrically placed with respect to the line. If then one of these is a straight line so will the other be. The surface will therefore, if it is a ruled one, be a quadric. But the ruled surface may be applicable on a surface of revolution without being a surface of revolution. We now inquire what property the ground form must have if it is to be applicable to a surface of revolution with generators corresponding to the meridian lines. Taking as directrix an orthogonal trajectory of the genera- tors we have e ^a = <J U 2 + (j; 2^2 + 2 tf u + i) <l v * m (1 OG . 5) If then this form is to be applicable to a surface of revolu- tion M and JVmust be constants, and we see that the ground form may be written rfs 2 = tin* + (u- + a 2 ) dv* ^ ( 1 06 . G) where a is a constant. Thus the catenoid and the helicoid will both have this form applicable to them. 107. Surfaces cutting at one angle all along a generator. We now wish to investigate the cor^dition that two ruled surfaces with a common generator may intersect at the same HO THE RULED SURFACE angle all along that generator. The condition will be found to have an interesting connexion with a particular class of congruences. We have seen that uM (cos \ITJJL + sin -fyv) -f sin 0/j, is a vector parallel to the normal to the ruled surface at the extremity of the vector 0. As we move along the generator this vector turns through an angle, remaining of course perpendicular to the generator. The vector product of the above vector and the neighbouring vector ( u + d u ) M (cos i//yz 4- sin tyf) -h sin 6/n is Mdiis'm 6 sin tyX. But the vector product is also (M 2 u* + 2 MM cos ^ sin + sin 2 0) </eA, where de is the angle turned through ; and therefore ^ _ _ -Wain SUM/T . (1071) du M-u* + 2 Ala sin flcos^-f H\I\* ^ ' ' Let M k sin sin \r where k is the ratio of the angle between two neighbouring generators to the shortest distance between them. Then dti (ku + cot ^)~ + 1 The equation of the line of striction is ku + cot \fs = 0. If therefore we measure u not from the directrix but from the line of striction we have the formula It follows that if we have two ruled surfaces, for which k is the same, and one of the surfaces is given a movement in space, bringing one of its generators into coincidence with CUTTING AT ONE ANGLE ALONG A GENERATOR 141 the corresponding generator of the other, and the correspond- ing points of the line of striction into coincidence, then the two ruled surfaces will intersect at the same angle all along that generator. 108. The ruled surfaces of an isotropic congruence. Let us now consider a ruled surface referred to its line of striction as directrix. Now ( 100) ' = A' = cos <9A-sin 6v. and, since the line of striction is the directrix, q is zero. We therefore have /A = 0. (108 . 1) Suppose now that is a vector depending on two para- meters u and v, and that A is a unit vector depending on the same two parameters. Consider the congruence z + w\. The congruence is said to be isotropic if and A correspond orthogonally. [See 81.] We have as the conditions for orthogonal correspondence <L A . = > <L^ + ^ 1 = ' Q*=> and therefore Aj = A^, A 2 = c/A^ 2 , where a is some scalar function of u and v. We thus have d\ - a\d, (108 . 2) whatever be the values of du and dv. The ruled surfaces of the congruence are obtained by con- necting u and v by some equation. For any ruled 'surface of the congruence wo therefore have f\ - 0, A = aA^ where the dot denotes differentiation along the arc of the curve chosen. This arc will be the line of striction since 142 THE RULED SURFACE and, since q is therefore zero, X = r cosec 0A It follows that a = r cosec 0, that is, from our definition of k ( 107), a = L (108.3) The ruled surfaces of the isotropic congruence therefore intersect at the same angle all along their common generator. They have all the same k at the point where the common generator intersects the surface w = 0, and their lines of striction all lie on this surface. This surface is the central surface of the congruence. CHAPTER VIII THE MINIMAL SURFACE 109. Formulae and a characteristic property. If we give to z y the vector which traces out any surface, a small arbitrary displacement normal to the surface at the extremity of 2, we have z' = z -f A , where is a small arbitrary parameter. Since z'^Zi + W + Xtl, s' a = 3 2 we have a' u = a n 2z l \ 1 t ) a' ]2 = a 12 22 1 A 2 , 22 that is, by (50 . 9), (109. 1) If the area of the surface is to be stationary, under this variation, then a must be stationary, where a = a 11 tf 22 -ai 2 , r since the area is a^dudv. j We therefore have an^22 + 22'ii- 2a iAa = 5 (109.2) that is, the sum of the principal radii of curvature must be zero. The surface of minimum area, the minimal surface as it is called, is therefore characterized by the property 22 / + Ji" = (109.3) where R' and R" are the principal radii of curvature. If we refer to lines of curvature as parametric lines on any surface *j = .R'A lf z 2 =#" t A 2 , and therefore, if cd 144 THE MINIMAL SURFACE is the ground form of the surface, the ground form of the spherical image will be <*udu* a Z2 dv* ( . WF (^")* The surface and its spherical image will therefore be similar at corresponding points if, and only if, (R")\ (109.5) that is, if the surface is a sphere or a minimal surface. On a minimal surface and therefore 2 'A 12 + R'^ -f R\\ 2 = 0. It follows that and therefore without loss of generality we njay say jR'Af=-l, Jf2 / Xi=-l. (109.6) The ground form may then bo taken as R'(du 2 + dv 2 ), (109.7) the asymptotic lines as d f u? dv 2 = 0, (109 . 8) and the ground form of the spherical image as (R')- l (du* + dv*). (109.9) Wo may now write R instead of R' t and since the ground form of a sphere of unit radius is we must have R (d 2 + sin 2 Od<f> z ) = dv? + dv 2 . (1 09 . 1 0) A If we take w = cot-e 1 ^, 2 we see that the complex variable w is the complex variable on the plane on to which the sphere of unit radius can be projected stereographically from the pole, if we take the pole as the origin from which 6 is measured and take the plane as the corresponding equator. * FORMULAE AND A CHARACTERISTIC PROPERTY 145 If w denotes the conjugate complex A we see that 4 sin 4 diu dw d& 1 -f sin 2 2 A and therefore 4 R sin 4 - div dw = cZu 2 -f (v 2 . (109. 11) If wo regard u + tv as the complex variable of another plane and denote it by x, we have 1 4 Now the curvature of the form 4 jfisiri 4 dvnlw 4 = cosec* is zero ; and, from the formula for the curvature of the ground form ( faa 2fdudv, we have f*K =/ lt / a ff l2 , and therefore ft sin 4 =f(w)F(<iv), (109. 12) Z where / and F are functional forms. If the surface is to be a real surface these forms must be conjugate forms. A Since cosec 2 1 -h ww the formula for R may be written R = ( 1+ ivwff (w) F (w). (109.13) We notice that in a minimal surface the asymptotic lines are perpendicular to one another in general though not necessarily so at a singular point. This property is character- istic of the minimal surface. 110. Keference to null lines. Stereographio projection. We now choose as the parametric lines on the minimal surface its null lines ( 45) and, instead' 1 of writing iv and w, we take u and v to represent tAese complex quantities. 2843 U 146 THE MINIMAL SURFACE "The spherical image will therefore also be referred to ifes null, lines and their parameters will be the same w and w or u and v. The normal to the surface is therefore given by (1 + uv) X = (u + v)t'-i(u-v)i"+(uv-l)i f ", * (110.1) where /, t", i" are three fixed unit vectors mutually at right angles and i denotes V 1. It is now convenient to introduce two vectors defined by . (110.2) These vectors are conjugate vectors and of course not real. They are, in fact, generators of the point sphere whose centre is the origin. Such point spheres must play in solid geometry the same part that the circular lines through a point play in plane geometry. We may easily verify the following relations between p, a, and X ; = t(\ + uv) <2 X, 2p = (l+iiv)*\ 2 , 2(7 = P\ = ip, CrX = i(T, pA x = \, crX 2 = ~fX, -2(X, P 2 = 0, cr 2 = 0. (110.3) We have seen ( 109) that the complex variable u on the sphere Q u = cot-e </> (110.4) is the complex variable on the equator when we take the ground form of the sphere to be and project the sphere, from the pole from which we measure d, stereographicaJly on to the equator. The conjugate corrfplpx v is the image of u in the real axis of the plane. < REFERENCE TO tfULL LINES 14? The complex u fixes a real point on the sphere, since wh&ri u is given its two parts are given and so its conjugate v is given. If Uj is the complex which fixes a point I\ on the sphere and u 2 is the complex which fixes the diametrically opposite point on the sphere, we have 1+1^2 = 0, (110.5) and consequently we also have We should notice that we cannot have uv +1 = 0. The complexes which correspond to the two opposite ends of a diameter may be called inverse complexes. 111. The vector of a null curve. A null curve is defined as a curve whose tangent at every point intersects the circle at infinity. Another way of stating the same definition is to say that the tangent at every point is a generator of the point sphere at the point. If z is the vector which traces out a null curve we therefore have Now the components of a vector which satisfies the equation X" = may be taken as proportional to and therefore we must have where/" (u) is some scalar function of the parameter u. It follows that the vector z which traces out a null curve may be defined by since the third derivative of p vanishes. We now denote the vector of the null curve by a, where a = pf" (u)- Pl /(u) + p u /(u). (111.3) 112. Self-conjugate null curves. They may be (1) uni- cursal, (2) algebraic. The conjugate nujl curve to a is clearly (112.1) 148 THE MINIMAL SURFACE where / is the conjugate function to /, and a- the vector we have defined in terms of its parameter v. A null curve is said to be self-conjugate, when for each value of u a value v' can be found, where v' is the conjugate complex to a complex u', such that We generally write p without specifying its parameter u, but sometimes we may need to bring the parameter into evidence and then we write it p u . Differentiating the equation <*u = <V we have pj'" (u) = <v/'" (v 9 )^, so that Vpu<r v f = > and therefore 1 + u v' = 0. (112.3) If we now write p for p u , and o- for ay, we "have p + 0u 2 = 0, PJ + 0-2+20-U = 0, tt 2 p n + (T 22 + 2ii(r 2 + 2u 2 (r = 0, and we can write = - <7U 2 /" () + K + 2<TW.)/' (It) -(<r. 2 + 2w<r 2 +2uV)^, (112.4) tv ^ = -/" ("') - ^/' (^) + ^ 2 /(^'). (112.5) If we now equate the coefficients of the vector cr^ on the two sides of the equation a =a, / we see that /(u) = - it 2 /( --- ) (112.6) and we see further that this single condition is sufficient to satisfy the equation # . ^ lt (112 7) tt In order then that a null curve may be self-conjugate it is necessary and sufficient that the function / which defines it should have the property (112.8) SELF- CON JUG ATE NULL CURVES 149 If we take 7 > = >- (112.9) where the coefficients are any real constants and the summa- tions may pass to any limits, we see that the function will satibfy the condition necessary to determine a self- conjugate null curve ; and we see that this is the most general function which will do so. If we only take a finite number of constants the self- conjugate null curve which results will be unicursal, More generally, if we take /(n) to be an algebraic function of u y then /' (u) and /" (u) will also be algebraic functions of u. We can then express the Cartesian coordinates of any point on the self- conjugate null curve rationally in terms of f( u )> /' ( u )> f" ( u ), and u. Wo shall then have six algebraic equations, connecting the three Cartesian coordinates and the four quantities /(u), /' (u), f" (u), and u. We can eliminate these four quantities and there will result two algebraic equations connecting the Cartesian coordinates. We have now seen how to construct null curves and self- conjugate null curves; and also how we can construct sell -conjugate null curves which will bo unicursal; and yet more generally how to construct self conjugate null curves which will be algebraic. 113. Generation of minimal surfaces irom null curves. Double minimal surfaces. When the minimal surface is referred to the null lines on it as parametric lines we have n = 0> o 22 = 0, and therefore, since a 11 /2 2:J -f-a 22 /} 11 = 2# 12 /2 12 , (113.1) we must have /2 12 equal to zei'o. 150 THE MINIMAL SURFACE That is, we have z\ = 0, z\ 0, z } A 2 = ^Aj = : and therefore, since \z l = 0, X0 2 = 0, we have Xz l2 = 0. We also have, from z\ cu = 0, that -1^12 = ^ ^2^12 == 0> and therefore 12 pA* where > is a scalar. But A: 12 = 0, and therefore z l2 = 0. (113.2) The minimal surface is therefore a particular case of a translation surface. A translation surface is defined by s = a + ft * (113.3) where a is a vector describing a curve whoso parameter is u and ft a vector describing a curve whose parameter is v. We see why it is called a translation surface as we can generate it by translating the u curve along the v curve or translating the v curve along the u curve. We might also define a translation surface by 20 = a + /3, (113.4) when we see that it is the locus of the middle points of chords one extremity of which lies on one curve and one on the other. In the case of the minimal surface we also have (cZa) 2 = and (d/3) 2 = 0, since z\ = and z\ = 0. The minimal surface is therefore given by where a and /3 are vectors tracing out null curves. If we confine ourselves to ral minimal surfaces the null GENERATION OF MINIMAL SURFACES 151 curves must be conjugate and the parameters of the two points must be conjugate complexes. It is obvious that such conjugate null curves will, if the corresponding parameters are conjugate complexes, give a real surface, and the converse may be proved. If the null curve is a self- conjugate curve, however, we must take as the corresponding complex, not the conjugate complex, but the inverse complex. Thus the general real minimal surface is given by 20 = <x IL + a v ] (113.5) and the real minimal surface generated from a self-conjugate null curve is given by 2s = a M +a/_i\, (113.6) \ ~ ?/ where the suffix is the parameter of the null curve which is to be taken. We notice that in the minimal surface 1Z = + /_ IN, as we pass from the point whose parameters are u, v by a continuous path to a point whose parameters are , we return to the point from which we started ; the z of the point will be the same but the X will be changed into X. That is, we are on the other side of the surface. For this reason the surface is called a double minimal surface. 114. Henneberg's surface. We have now seen how minimal surfaces are generated from null curves, and how real minimal surfaces are to be obtained, and how i;eal double minimal surfaces may be generated. From what we said about the construction of null curves we see how to obtain minimal surfaces which will be rational functions of their parameters and how to obtain more generally algebraic minimal surfaces; and from what we said about the construction of self-conjugate null <jui*ves we can construct these surfaces to be double minimal surfaces. 152 THE MINIMAL SURFACE Thus will be an example of a real double minimal surface as may easily be verified. It is known as Henneberg's surface. It may easily be shown that a minimal surface will then only be algebraic when the null curves which generate it are algebraic. 115. Lmes of curvature and asymptotic lines on minimal surfaces. We have for a minimal surface and, if the surface is to be real, 2s = It follows that and therefore 4 -^ = /'" (it)/"' (v) pJr. But 4/^ = (l+'uyJ'ATX,, and, if R, Jl are the principal radii of curvature of the surface, ^ _ ,> 2 r\ ^1^2 ~ ~^ A 1 A 2' We therefore have U3R*-=f"(u,)f'"(v)(l+uu)*. (115.1) then 4^=(l 4 t,), , o and 2s = THE LINES OF CUKVATUKE AND ASYMPTOTIC LINES 153 We then have so that if i//- ~ f- irj, \fr ?; we come back to the ground form ds* = R(d* + dri*) for the surface. The lines of curvature are = constant, ?/ = constant, and the asymptotic lines are -f 77 constant, g rj = constant. We may therefore say, if R(f>(u) denotes the real part of (it), that one family of the linos of curvature is R f Vf" (u) du = constant, (115.2) j r __ and the other Ri </f" (u)du constant ; (115.3) j whilst the asymptotic lines are given by R f V7f tTr (^) du = constant, (115.4) R (V-/'"(tf)<Ztt = constant. (115.5) 110. Associate and adjoint minimal surfaces. The surface obtained by substituting for / the function e l( *f wjiorc a is a real constant is said to be an associate minimal surface to /; and when we take for a the number ~ it is said to be the adjoint minimal surface. An associate minimal surface is applicable on tho surface to which it is associate and the gormals are parallel at corresponding points. 154 THE MINIMAL SURFACE If { is the vector which traces out the adjoint surface to z 2<b :-/'" (U)pdtl+f" (V) (Tttv, 2df = L (/'" (u) pdu-f" (v) <rdv), (1 16 . 1) so that these two surfaces will also correspond orthogonally. We see that z i traces out not a surface but a null curve, and z + i traces out the conjugate null curve. Since p\ //>, crA JCT, we also see that d{ = Xth. (116.2) If then we are given a curve on the surface we shall know the which will correspond to z along this curve, if we know the normal to the surface along the curve. We shall therefore know c-f/^ and z i along the given curve, and thus have the null curves which generate the minimal surface. p ^ ^ The formula Oi (l -^-i\ \dz % (116.3) j is due to Schwartz. CHAPTER IX THE PROBLEM OF PLATEAU AND CONFORMAL REPRESENTATION 117. The minimal surface with a given closed boundary. Any account of minimal surfaces would be incomplete without some reference to the problem proposed by Lagrange : ' To determine the minimal surface with a given closed boundary, and with no singularity on the surface within the boundary/ This problem is known as the Problem of Plateau, who solved it experimentally. The problem has not yet been solved mathematically in its general form ; but has been solved in some particular cases, where the bounding curve consists of straight lines and plane arcs of curves. Consider a part of the bounding curve, which is a straight line, on a minimal surface. This line must be an asymptotic line on the surface. Now we saw ( 109) that, when the surface is referred to the lines of curvature, as parametric lines, the equation of the asymptotic lines is dtf-dv* = 0; 017.1) and the ground form of the surface is R(<lu* + dv*)> (117.2) and the ground form of the spherical image is R- l (d>u? + dv*). (117.3) Wo conclude that when the surface is conforinally repre- sented on the plane, on which u and v are the rectangular coordinates, the asymptotic lines are conformally represented by lines parallel to the bisectors of the angle between the axes, and the lines of curvature, and also their spherical images, are conformally represented by lines parallel to the axes. * 156 PLATEAU AND CONFORMAL REPRESENTATION If a part of the bounding curve is a plane curve, whose plane cuts the minimal surface orthogonally, and is therefore a geodesic, it must be a line of curvature. It will therefore be conforrnally represented on the plane by a line parallel to one of the axes. If then the whole of the bounding curve is composed of straight lines and such curves, the bounding curve will be conform ally represented on the plane by a figure, bounded by straight lines, parallel either to the axes or to the bisectors ; and the part of the minimal surface, within the boundary, will be represented by the area of the plane within the polygon. Next let us consider the spherical image of the surface within and on the boundary. At each point of the boundary, the normal to the surface will be perpendicular to a direction, which will not change as we pass along a continuous part of the boundary, but will change at each angle of the boundary. The boundary will therefore consist of * arcs of great circles. If therefore we can find a function of ^v 9 the complex variable which defines the position of any point on the sphere, which will transfoim the spherical boundary into the plane boundary, and points within the spherical boundary to points within the plane boundary, we shall have u-t iv known in terms of w, and can proceed to find the required surface as follows. We have (109 . 10) for an element of the sphere da* = dff* + sin 2 6d$\ (U 7 . 4) so that w = cotf^ (117.5) j is the complex variable which defines the position of points on the sphere. The normal, to the sphere, which is given by w, is by (110 . 1) (l+ww)\ = (w + w)i' i(w w)i" + (ww-~l)i" f , (117.6) where i' t i"> i" are* fixed unit vectors, mutually at right angles, MS \/ 1, and w the conjugate complex to w. MINIMAL SUEFACE WITH GIVEN BOUNDARY 157 Now we know that in terms of u and v and therefore R~ l (du? + dv*) = dO* + *iri 2 6d$*. (117.7) As we have seen in 109, R is therefore known, being given by dx (117.8) dw We can therefore construct the surface since R and A are known in terms of w and w . We can retrace our steps and see that the surface we have obtained satisfies the conditions required. We are thus led to the problem of con formal representation, and this we proceed to discuss, so far as it bears on the question before us. 118. The notation of a linear differential equation of the second order with three singularities. Let a, c, b be three real quantities in ascending order of magnitude, and let x be a complex variable. When x lies on the real axis between oo and a, or between b and + oo , we see that b G j' a b a x c lies between zero and positive unity. When x lies between a and c c b x a c ct x - b lies between zero and positive unity. We also see that the reciprocals of these two expressions lie between zero and positive unity, when x lies between c and b. When x is complex we see that the modulus of one of the first two expressions is less than unity, or the modulus of each of the reciprocals is less than unity. Let a lf Of 2 , /?!, j8 2 , y 19 y a be six quantities real or complex, but such that ^ + ^ + ^ + ^4.^ + ^ = i ; (118.1) and such that the real parts of a 2 - a i> ft-01. ,72-71 are each positive. 158 PLATEAU AND CONFORMAL REPRESENTATION Let a? a (za)(x-l) (x c)Q z+'-y'-y*. (1,8.2) b x c T - - 1 - - j x a xb x c (118.3) and let (a t , ft, y u 2 , ft, y a , a, 6, c, a) denote the hypergeometric series where p^^ + ft + yx, ? = l + ft + y l , r = 6 ic a __ * a x b We notice that P and Q are unaltered by the following substitutions : focg, (ft/U ( yi y a ), (yaai)(y 2 a 2 )M, (,^i) (*&)(&) (118.5) 110. Conformal representation on a triangular area. Consider now the diffurential equation It is known, and may easily be proved, that is a power series, beginning with (x a)* 1 for its first term and expansible in powers of x a in the neighbourhood of x = a, which will satisfy the differential equation. This series when x lies on the real axis is valid when x lies between a and c. It is therefore valid at any point in the plane, the circle through which, having a and 6 as limiting points, intersects the real axis between a and c. CONFORMAL REPRESENTATION ON A TRIANGULAR AREA 159 Another power series also beginning with (x a)* 1 can be obtained from the first by applying the substitution to it. The two series will therefore be identical at any point where they are both valid. The second is valid for real values of x between oo and a arid between b and -f oo. The region for which it is valid, when x is complex, can be obtained by a similar rule to that which was used as regards the first series. When one series is valid, but not the other, the valid series is a continuation of the other. We denote these series by Fa r l>y applying the substitution (oqaj we obtain two other series beginning with (x a)S valid over the same part of the plane. We denote these series by Fo^. By applying the substitution (y^) (y 2 J (cu) we get two series, Yy l beginning with (x c) 71 , and Fy 2 beginning with (x c') 7z , valid over the part of the plane which corresponds to real values of x between c and b. By applying the substitution (Piyi)(P 2 y 2 ) (be) to these last two series we get two other series, F/Jj beginning with (x 6)^, and F/8 2 beginning with (x b)^ 2 , valid over the same part of the plane. All these series, when valid, satisfy the equation. r . Ya 2 Let w -fr-^ Y*i Then we see that, as x describes the real axis from 6 to -f oo , and then from oo to a, w varies continuously and its argu- ment is TT (a 2 (Xj), if we agree that the argument of a positive quantity is to be taken as zero, and the argument of a negative quantity as ?r, as x describes the real axis in this definite way. t * Let Q be the point in the'iu plane which corresponds to 160 PLATEAU AND CONFORMAL REPRESENTATION x = 6, and let P be the origin in the w plane corresponding to x = a. As x describes the path defined, w describes the straight line QP. When x describes the semicircle about (6 the argument of w diminishes by 7r(a 2 o^), and as x describes the real part of the axis the argument of w remains zero, till we come to R, which corresponds to x = c. Wo must now consider what happens as x describes the semicircle round c, and then, passing along the real axis, comes to b and passes round the semicircle there. Over any part of the plane which corresponds to real values of x between c and b we can express w in either of the forms C + D- where A t B, C y D and A', B', (7, D' are certain constants. We see this from the known properties of a linear differential equation of the second order. Now the argument of ~^ is the same as that of * Yi /X C, A7a-7i ( -(c-a)) \X--(t '/ and therefore zero, as x passes iflong the real axis from c to b. CONFORMAL REPRESENTATION ON A TRIANGULAR AREA 161 It follows that w describes a circle which passes through Q and E. ^ The increment of the argument of w ^ as we pass along C the semicircle c is the same as the increment of the argument of y^ ; that is, it diminishes by ^(y^ yi)- The circular arc through R therefore makes an angle 7r(y 2 "~yi) with jRP. In the same way we see that the angle at Q is 7r(/3 2 jSJ. Since, when x moves from its real axis to the positive side of its plane, w must move to the inner part of the triangle PQR y we see that the positive part of the plane of x is con- formally represented by the inner part of the triangle. 12O. The w-plane or part of it covered with curvilinear triangles. Consider now the transformation x , ^px + q rx + s where p, q, r, 8 are any constants, real or complex. If x describes a circle (or as a particular case a straight line) in its plane, so will x'. If x l and x^ are any two points inverse to the circle #, then x\ and o/ 2 will be inverse to the circle x'. We thus see that if P, Q, R are the three points which, in the above transformation, with F 2 ~ Fo^ substituted for x, correspond to the singularities at a, 6, c, the curvilinear triangle PQR, formed by three circular arcs intersecting at angles ATT, /i?r, VTT, where * A = ,,-!, f= 0S-&, '"=72-71. 2843 y 162 PLATEAU AND CONFOKMAL REPRESENTATION will enclose the part of the w plane, which conformally repre- sents the upper part of the x plane. Let w be the complex variable which defines any point 8 within tbe triangle PQR, and let w l be the complex variable which defines the point 8 19 which is inverse to $ with respect to the arc RQ. Let rf be the substitution which transforms the arc RQ to a part of the real axis of w in its plane. Then and (120.2) rw + s m\ + s ' arc inverse to one another with respect to the real axis of iv. Let f(x) be the function of x which we found would in this case transform the upper part of the x plane to within the curvilinear triangle in the w plane. We now assume the quantities ot 19 a 2 , J3 19 /? 2 , y T , y 2 to be all real. Along the real axis of the plane x the coefficients in /(*') will be real, and therefore f(x) will be the function which will transform the lower part of the plane x to points without the curvilinear triangle, where x denotes the conjugate variable to x. We therefore have It follows that w = - and w = - . (1 20 . 3 ) p-rf(x) l p-rj(x) 9 v ; and consequently we have p-rf(x) Eliminating f(x) we have * (ps qr)w + 08-^ qs W =! * / _ 1_. ( jjw rp) w + p$ qr A DERIVED NET- WORK OF TRIANGLES 163 If then w = F(x), and w l </) (x), then (x) = (fo-g*)g(g) + g'-gg . (120 . 5) v ' If JFJ is the inverse of P in the arc Q-R, we thus see how the lower part of the x plane is conformally represented on the triangle P$R in the w plane. Similarly if (^ is the inverse of Q in RP, and 7? x the inverse of R in PQ, we can conformally represent the lower part of the x plane on the triangle ^.RP, and on the triangle R.PQ. Just in the same way from the triangle 1\QR we can by inversion obtain three other triangles, one of which will be the triangle PQR. These triangles will give conformal repre- sentations of the upper part of the plane x on the plane of w. Proceeding thus we cover the whole, or a part, of the ^u plane with curvilinear triangles. 121. Consideration of the case when triangles do not overlap. In general these triangles will overlap, so that a point in the w plane may be counted many times over : in fact, unless A, //, v are commensurable, a point in the w plane which lies within any triangle will lie within an infinite number of triangles. If, however, X, /z, v are each the reciprocal of a whole number there will be no overlapping at all. We now confine ourselves to this <!ase. 164 PLATEAU AND CONFORMAL REPRESENTATION One and only one circle can be drawn to cut orthogonally the arcs of the fundamental curvilinear triangle in the w plane. By inversion we may take PQ and PR to be straight lines. We see that the two straight lines and the circle divide the w plane into eight parts. We see, however, by considering the original figure with which we began this discussion, that the triangle with which we are concerned is the shaded one. For at the point L the variable w will move in the direction of the arrow, for a corresponding movement of z to the upper part of the x plane ; and, as w will not move oft' to infinity, the triangle could not be the outward part of The triangle PQll is therefore of one of the two forms In case (1) P must lie within the circle of which EQ is the arc. THE CASE WHEN TRIANGLES DO NOT OVERLAP 165 For, otherwise, the sum of the angles at Q and R being for we are now assuming A = - , /* = -, ^ = ^ /I 1\ \q r/' the sum of the angles at Q' and R' would be ,(2-1-1); \ q r/ and therefore 2 < 1. q r But this is not possible if q and r are integers. No real circle can therefore be drawn with P as centre to cut the arc QR orthogonally in case (1). The two cases are therefore thus distinguished : in case (1) A + /* + i/>l, (121.1) and the orthogonal circle is imaginary : in case (2) X + ^ + ,/< l, (121.2) and the circle which is orthogonal to the three arcs is real. 122. Case of a real orthogonal circle as natural boundary. Taking case (2), the circle, whose centre is at P and which cuts the arc QR orthogonally, must intersect the circle QR at the points of contact of tangents to the circle from P. Clearly these points are without the arc QR, since the arc QR is convex with respect to P. The points P> Q, R therefore lie within the orthogonal circle. When we invert with respect to a point outside the orthogonal circle we have three circular arcs within the new orthogonal circle*. By considering the point P l which is the inverse bf P with respect to QR, we see 166 PLATEAU AND CONFORM AL REPRESENTATION that P l also lies within the orthogonal circle. Proceeding thus we see that all the curvilinear triangles are within the real orthogonal circle which corresponds to the case A+ fJL + V < 1. In this case, therefore, only the part of the w plane which lies within the orthogonal circle is covered with the curvilinear triangles, which conformally represent the x plane on the w plane. This circle is therefore the natural boundary of the function which, with its various continuations across the real axis of x, conformally transforms the x plane to the w plane. Since there are an infinite number of solutions of the in- equality 1 i i - + + - < 1, P 9 r where p, q, and r are integers, we get an infinite number of triangles which grow smaller and smaller as we continue to invert and invert : and as we approach the -boundary the orthogonal circle the triangles tend to become mere point triangles. 123. Fundamental spherical triangles when there is no natural boundary. We now consider the first case when 1 1 1 -+ + > 1 p q r and the orthogonal circle is imaginary. If we stereographically project the w plane on to a sphere which touches the w plane at the real centre of the orthogonal circle, the fundamental curvilinear triangle becomes a spherical triangle which we shall now denote by ABC. The only possible solutions of the inequality are (1) p = 2, q = 2, r = m; (2) p = 2, q = 3, r = 3 ; (3) p = 2, q = 3, r = 4 ; (4) p = 2, q = 3, r = 5 ; or equivalent results obtained by permutation of the integers. We lose nothing by taking A, B, to be the correspondents to the singular points a, c, b in the x plane. We may thus have *f or the fundamental spherical triangle any of the four figures which follow. FUNDAMENTAL SPHERICAL TRIANGLES 167 The operation of inversion is now replaced by the simple operation of taking the reflexion of each vertex with respect to the opposite side. We see at once that the whole surface of the w sphere is covered by the triangles and their images. In the first case we have 2m triangles in the upper part of the hemisphere and 2 m triangles in the lower part. In case two we have a triangle whose area is ^ that of the sphere, and by taking the six triangles with a common vertex at A we have an equilateral triangle whose area is that of the sphere : that is, we have the face of a regular tetrahedron. In case three, which is just that of the triangle formed by bisecting the angle G in case two, we have a triangle whose area that of the sphere. By taking the eight triangles with s a common vertex at A we have the equilateral quadrilateral whose area is ^ that of the sphere, that is, the face of a regular cube. Its angles are each ,- ; and it is also the figure o 168 PLATEAU AND CONFORMAL REPRESENTATION formed by planes, through the centre of the sphere circum- scribing a regular tetrahedron, perpendicular to two pairs of opposite edges. In case four we have a triangle whose area is T ^ that of the sphere. By taking the six triangles, with a common vertex at B, we obtain an equilateral triangle, whose area is ,jV that of the sphere : that is, a face of the regular icosahedron. 124. Summary of conclusions. When A, /z, and v arc then the reciprocals of integers, we have found functions w of the complex variable, which will conformally transform the upper and lower halves of the x plane into the area within the curvilinear triangles in the \o plane. To each point in the x plane there will correspond, in the w plane, one point in each triangle or in the triangle adjacent which is its inverse. The real axis will be transformed into the circular boundaries of these triangles. Two different points in the x plane cannot h&ve the same w to correspond to them. For by taking A, /x, and v to be the reciprocals of integers we have provided against any over- lapping in the w plane. It follows that # is a uniform function of iv. In the case where A -f /z -f- v > 1 there are only a finite number of values of w which will make x zero or infinite ; and therefore x will be a rational function of w. We could express each value of w which makes x zero in terms of any one, and thus obtain the numerator of the rational function. Similarly we could find the denominator. As we only wish to give a general explanation we do not enter into any details. We have now shown how to represent the 10 plane, or its equivalent sphere, on the x plane. 125. Representation of the x-plane on a given polygon. To complete the problem of conformal representation in so far as it bears on the problem of Plateau, we have now only to show how the x plane can be conformally represented on a given polygon. The' procedure is much the same as in the problem we have just discussed, tout much simpler. REPRESENTATION OF OHPLANE ON A GIVEN POLYGON 169 Let a, o, b be defined as earlier and let a, f}, y be three real constants which are positive, and such that a + /J + y = l. (125.1) Let X = I** (x-a)"- 1 (x-l)f- 1 (x-c)^ 1 dx, (125 . 2) J-oo and let A be the position which X attains as x moving along its real axis approaches a. As x moves along the real axis in its plane from oo to a, the argument of X is zero, so that it too moves along the real axis of its plane. As x moves along the small semi- circle with centre at a, the argument of X diminishes by OLTT. As x then moves along the real axis to c, X moves along a straight line AC to (7, the point which corresponds to c. When x describes the semicircle at c, the argument of X again diminishes by yir. Then as x moves along the real axis from c to 6, X describes a straight line CB to B the point which corresponds to b. X is now again on its real axis ; and as x describes the semicircle at b the argument of X diminishes by /JTT. Finally as x moves along the real axis to +00 and then from GO to a, X describes the straight line BA. We thus have the figure in the plane of X, and the upper half of the x plane is con- formally represented by the area within this triangle. By a transformation of the form X' = pX 4- q where p and q are constants the triangle may be transformed into any similar and similarly placed triangle in the plane of X ; and thus the upper half of the x plane may be conformally represented by the area within the triangle ABO which lies in the piano pf X anywhere. * 170 PLATEAU AND CONFORMAL REPRESENTATION We thus see, as before, that the pland'of x can be repre- sented by a series of triangles in the plane of A", which will cover it completely. But if there is to be no overlapping we must have a, /?, and y to be the reciprocals of integers. These integers must satisfy the equation 1 - q 1 - r (125.3) and we see that the only solutions of this equation are p = 6, 3=3, r = 2 ; p = 4, 3 = 4, r = 2 ; j> = 3, 3=3, r=3. (125.4) Wo thus have three cases and we see into what kind of triangles the given polygon must be decomposable in order that x may be a uniform function of X. We see that X is a doubly periodic function of x ; and from the above triangles, and their images in the sides, with respect to the opposite vertex, we can construct the period parallelograms. 126. We have found corresponding to each value of ttf, the complex variable of the sphere, a definite value of x. This value of x will under certain circumstances which we have considered be a rational function of w. To this value of x we must choose/as its correspondent X, that value, or those values, which lie within the given polygon. Since REPRESENTATION ON A POLYGON 171 the values of w which lie on th# boundary of the spherical polygon are to correspond to values of X lying on the boundary of the plane polygon, and since these values of w correspond to points on the real 'ftxis of x, we see that the polygon must have its boundary made up of sides of the elementary triangles in the X plane. The principal results in the theory I have tried to explain in outline are due to Riemann, to Weierstrass, and to Schwartz; and my presentation is based on the treatises of Darboux and Bianchi. The connexion of this branch of Geometry with the Theory of Functions is interesting. CHAPTER X OKTHOGONAL SURFACES 127. A certain partial differential equation of the third order. We now want to consider the theory of a triply infinite system of mutually orthogonal surfaces ; and \ve begin by considering the partial differential equation of the third order p -f q Y- + sech 2 x = q tanh x, (127.1) where 20 = tan- _ (127 . 2) v ' and z is the dependent variable and x, y, and w the inde- pendent variables. [Here p, q, r, s, t denote respectively We shall see that it is on this equation that we depend when we wish to obtain the general system of orthogonal surfaces. Let ~ be any function which satisfies this equation, and let -, cx cy d W ~ p cosh 2 x - h Q cosh 2 x - h - 1 dx * <)y ^w then it is not difficult to verify that VW- WV = F(pcosh 2 o;) . V. (127 . 3) It follows that a function u exists which is annihilated by the operators F and' W 9 and also a function v annihilated by U and W. A PARTIAL DIFFERENTIAL EQUATION OF ORDER 3 173 Wo may therefore regard x and y as functions of u, v, and w, and we have <$u <>u <H<> <>u <*u , a* + -, y. 2 = 0, r #0 + r y ., + r = 0, 2 3 t/0 where the suffixes 1, 2, 3 respectively denote differentiation with respect to u, v, w. But from the definition of 11 and v we have ^^6 J . c)?C _ c^U . c)^6 <)iC r + tan 6 . = 0, p cosh 2 a? r + q cosh 2 tc - + -- = 0, ^)ic ^y ^x <>y div ^v , A<*V ^ o ^v . ^v ^v . - cot ^ x - = 0, w cosh 2 x - + a cosh 2 x ^ + - = ; ex dy L dx J dy %w and therefore it follows that = iV 2 ~" 2/2 co ^ ^ = 0, = 0, 2/3 <1 cosh 2 ^ = 0. (127 . 4) We now see that so that the equation with which we began becomes 3 = ^ sinh a; cosh <r, (127 . 5) that is, 3 = 7/ 3 tanh x. We thus have the three equations #! + j^ tan = 0, .T 2 2/2 cot = 0, 3 = ?y 3 tanh oj. (127.6) Now let sin* , rj = e~~ ., - y-6' fc ~ " 2 ' cos ------ cos --r 2 A so that tan 6 = ^ , then we can verify that ** * * ' (127.7) 171 ORTHOGONAL SURFACES 128. A solution led to when functions satisfying a set of three equations are known. By retracing our steps wo may verify that if we have any three functions which satisfy these equations we shall be led back to the solution of the equation of the third order. For we have clearly x l + y l tan = 0, a* 2 ?/ 2 cot = 0, 3 = y 3 tanh x ; and therefore = a 1 , 17, c- = # V. ^u L <)v N OW A = S i n 2 Q u+ cos 2 6V, ~ = sin 6 cos 6 (V- U) ; and therefore we can verify that A ^ = _^*_ , <>2/ cosh 2 ^ <)aj cosh 2 x so that # 3 = > cosh 2 #, 2/a = 5 cosh 2 .r. (128.1) From # 13 -f 2/13 tan 6 + y l sec 2 0^ = we verify that U (p cosh 2 s) + tan CT (5 cosh 2 a) = (cot -f- tan $) q sinh x f cosh r/:, and therefore tan 2 5 = 28 + 2 7 tanha! . (128.2) r ^ + 2 ^> tanh a; Finally we see that --- = TF, oiy and thus the equation 3 = y 3 tanh x or 3 = ^ sinh a? cosh a/ becomes Wd = g sinh a? cosh x. (128.3) The equations are thus connected in the way we have described with the partial differential equation of the third order. 129. The vector qaq~ l , where a is a vector and q a quaternion. We now pass on to the geometry which we associate with these three equations. If we are given any vector an& any scalar quantity we can A ROTATED VECTOR 175 take the vector to bo r sin . e and the scalar quantity to be r cos 0, where is a unit vector. Let q = r (cos + e sin 0), (129.1) then q is called a quaternion, e is called the axis of the quaternion, 6 is called its argument, and r is called its modulus. A quaternion is thus just the ordinary complex variable of the plane perpendicular to the axis of the quaternion. We have q~ l = r~ l (cos 0-esin d). (129.2) Any other vector may be written xt' + ye, (129.3) where x and y are scalars and e' is some unit vector at right angles to e. We see that q(xt' + ye)q~ l is equal to x(e' cos 20-f e" sin 26) + yt, (129. 4) where e" is the unit vector perpendicular to 6 and e'. That is, if a is any vector, q(X q-i ( 12 g . 5) is just the vector oc rotated about the axis of the quaternion through an angle double the argument of the quaternion. 130. Passage from set to set of three orthogonal vectors. Let us now consider the quaternion 7= l+i + i)j + (k, (130.1) where i, j, k are fixed unit vectors at right angles to one another and , 17, f are any three scalar functions of the parameters u, v, and w. Dq~ l = l-i-r,j-(k, where D = 1 + 2 + rf + * . . (130.2) Let X = qiq~ l , /JL = qjq~ l , v = qkq~ l , (130 . 3) then A, //, v will also be three unit vectors mutually at right angles to one another, no longer fixed vectors but depending on the parameters u, v 9 w. Any system of mutually orthogonal unit vectors can be so defined. * 176 ORTHOGONAL SURFACES We easily see that (130.4) Now , WVir^tfiT 1 , and therefore W 1 !?!?" 1 = 9V/ "^ It follows that * + Oh - tf i (130.5) From qi ~ \q we have 7V/~ 1 ^""^ r /i7~ 1 := ^11 and therefore f/i'/" 1 ^ ^ r /V/~ 1 = ^r It follows that, since /zi> VJJL = 2A, z/X At/ = 2/z, A/z /x\ = 2y, Z)A 1 =-2( 7l -^ 1 + ^ 1 ) J , + 2(^-^ 1 +-^ 1 )^ (130.6) Let /S- 4 Zty' " = | 3 - itf, + ^ a) 4 D?'" = 7,3 - then we have proved (130.6) that X, = pr' vq'; and similarly we prove the other equations of the system (130.7) It may be noticed that the q', q", q'" as here defined have no direct relationship to the quaternion q where q= If >' = SETS OF ORTHOGONAL VECTORS 177 we see that A 1 = co / X, ^ = o>'/*> i/ 1 = a/j/, A 2 = oPA, // 2 = o>'>, i/ 2 = o)" v, A 3 = o/"A, ^ 3 ==o/ 7 >, j/ 3 = G>"V (130.8) and we can easily verify from the formulae given that a>' = 2, / 7?- 1 , a>" = 2<M- 1 , '" = 2 (M" 1 (130.9) where g is the quaternion. We also have as we proved earlier [see 90] the equations o)'" 2 a>" ;J o0"o>'", 0/30)'"! = &"'<*>'> 3>"i fi/2 == ^'^ (130.10) The angular displacements of the vectors >, /*, *> regarded as a rigid body are ^^ ,"<&,, n'" c l w . (130 . 11) 131. Rotation functions. So far we have been consider- ing a system of three unit orthogonal vectors of the most general kind depending on three parameters, and we have seen how they depend on the quaternion l+i--iM + fc. We now want to consider the particular system characterized by the property that p' q" = T f " = Q, (131.1) that is, by the property that , 77, satisfy the three equations (127 . 7) which in 128 we connected with the partial differential equation of the third order with which we began our discussion. We now have from (130 . 7) \ l = /ir'-rj', A 2 = fir", A 3 = -vq"', K = -Xr', fr = n >"-\r", to = n>'"> v, = X 9 ', ^ 2 = -w", v. = A^'- W /". It will be convenient to write when the above equations become = 0, /* 2 + i/(32)+X(12) = 0, (131.2) 178 ORTHOGONAL SURFACES The six functions (23), (32),- (31), (13), (12), (21) we shall call rotation functions. They are connected by the laws (21) a +(3l)(32) = 0, (23) 1 =(21>(13), (31) 2 = (32) (21), (12) 3 = (13) (32), (32), = (31) (12), (13) 2 =(12)(23), (21). = (23) (31), (131.3) as we can at once verify from the equations satisfied by A, fi y v. We can express these rotation functions, as we have done, in terms of , 77, and their derivatives. 132. A vector which traces out a triply orthogonal system. Now consider the system of equations (132.1) where a, /?, y are scalars to be determined by these equations. We see at once from the set of conditions (23), =(21) (13), (31) a =(32)(21), (12) 3 = (13) (32), (32), = (31) (12), (13) 2 = (12) (23), (21). = (23) (31) that they are consistent. Let a, ft y be any three functions which satisfy them, and let 2 = aA + jfy + yy. (132. 2) We have z l = ( 1 + j8(21)-f y(31))A, (132.3) and therefore the vector z traces out a triply orthogonal system of surfaces. Conversely we see that there is no triply orthogonal system of surfaces which cannot be obtained by this method. LINES AND MEASURES OF CURVATURE 179 133. Lines and measures of curvature. If we take ), 6 =0 2 + y (32) + wo have z l = a\, z. 2 = bp, z. A = cv, and we see that , = 6(21), 6 3 = c(32), c 1 = 3), (133.1) and that these three last equations together with a = -3u, 6= C<2 (133.2) (13) ' (23) * ' are equivalent with the first six. The three orthogonal surfaces are u = constant, v = constant, w = constant ; the unit vectors parallel to the normals at the extremity of the vector z are respectively A, ^, v. We have ^ = ^> ^ = ( 133 - 3 ) and therefore the curves along which only v and iu respectively vary are the lines of curvature on the surface u = constant, and its principal radii of curvature are (133<4) We thus have the fundamental theorem about lines of curvature of orthogonal surfaces, viz. that they are the lines in which the two other surfaces intersect one of the surfaces. If we consider the curve in space % along which only u varies, and if we suppose its principal normal to make an angle 6' with the vector /*, *and p' and cr' to be its two 180 ORTHOGONAL SURFACES curvatures, we have in the notation we used in considering curves in space (see 94) cos0' sin 6' (133.5) Now aX = X x , a/i = // 1 , av = j^, since acZu is the element of arc of the curve, and as we have we must have , (31) = "+ > = 0, (T Thus considering the three curves we get 6'+ i = 0, " (23, _ ^ ' "" (32) = "' + r,= 0, or (133.6) and we thus see another interpretation of the rotation functions. In the figure here given A, B, C represent the points where e" " A the vectors X, /*, ^ intersect the unit sphere whose centre is the origin ; that is, the points* where parallels to the three MEASURES OF CURVATURE 181 tangents to the curves intersect the sphere ; and A', E\ C' the points where the parallels to the corresponding principal normals intersect the sphere. The principal radii of curvature of the surface u = constant were, we saw, /> c (72) and (is)' that is, -p'" and therefore the measure of curvature is , / ' / 7) ///// - - //"/// P P P P Similarly we have - CO * C ' A ' '" - CosA ' B ' K" - ~ p'"p' Again, from the formulae 6'+- f = 0, d"+4 = 0, d'"+-*,, = (133.9) 0" CT CT we at once see that, if a line of curvature is a plane curve, its plane cuts the surface at the same angle all along it. 134. Linear equations on whose solution depends that of the equation of the third order. We now return to the equation of the third order (127 . 1), 12 * d 12 ^ ^ p coslr x + q cosh 2 x + ~ = q smh x cosh a?, dsc 2 ty 1*10 * ' where 2 5 = tan^ 2g+2 ^ tanh . r t + 2p tanh a; Suppose that z is any integral of this equation : we may suppose it expressed in the series * =/( 20 + ^0(0, 2/)+^ 2 ^(, 2/)4- ..., (134 . 1) and if the integral is a general one we may take / to be any arbitrary function of x and y. We shall show how wh8n / is a known function the 182 ORTHOGONAL SURFACES function <f> depends for its determination on a differential equation of the second order. Let and let /2 = cosh 2 x ~- ~ 4- cosh 2 # ~ ox ex ^y cy The equation which determines is then Qf2P-Pfi,Q-2 sinhacosh x (P 2 + Q 2 ) (134.2) Now let tv = W' + ^'Q, where w is a small constant whose square may be neglected, then s = /+W' + w'(0 + 2' ^r) + ..., (134.3) and by solving a similar equation to the above with /-M' f () </> substituted for /we should find + 2?^, (134.4) and thus obtain ty. Proceeding thus we see the system of linear partial differential equations on whose solution we depend for obtaining the coefficients of the different powers of w in the series for z. A particular solution of the equation of the third order would bo obtained by taking/ to satisfy the equation QflP-PflQ = 2 ^ sinh x cosh x (P* + Q*)> (1 34 . 5) ij when we could take to be equal to /. 135. Synopsis of the general argument. It may be useful at this stage to give a re8um6 of the general argument. z is a function of x, y, and w which satisfies the equation <>0 ^0 10 ^6 p- h q rr- V secrr# - = q tanh x, 1 * SYNOPSIS OF THE GENERAL ARGUMENT 183 , o/j x i where 2 6 = tan"" 1 A r t + 2 p tanh x W = 2> cosh- x ^~- + q cosh 2 a? + ^- ; * d# 2 c)7/ aw and ?6 and v are defined by Vn = 0, T^ = ; L"y = 0, Wv = 0. We can now express a, ?/, and in terms of if, v, and ?; ; and having done so we define , 77, by y-f<9 . + 6 cos - sin ~ __ _ _ 7/-fl' - ?/-g' - 2 ' cos cos ^ ^j 2i and we have The functions ^, 77, now define a quaternion '/= 1+^-f W + ^% where /, j, k are any fixed unit vectors at right angles to one another. Three unit vectors mutually at right angles are now de- fined by x = <]iq~ l , /* = </^/" 1 , v qhf- 1 , where D([~ l = lirij k and 7)= i+* +1? 2 + ^ These vectors are not fixed. We have (31) = 0, //> + */ (32) + A (12) = 5 and thus the six rotation functions (23), (32), (31)* (13),' (12), (21) 184 ORTHOGONAL SURFACES are defined. These functions satisfy the conditions (31) 8 +(I3) 1 + (23)(21) = 0, (31)(32) = 0, ^ = (21) (13), (31), = (32) (21), (12) 3 = (13) (32), (32), = (31) (12), (13) 2 = (12) (23), (21) 3 = (23) (31). The vectors X, /*, v are parallel to the normals at the ex- tremity of some vector z depending on three parameters which traces out the three orthogonal surfaces n = constant, v = constant, w constant. This vector z is defined by z = X + /3/i + yv, where a, j8, y are scalars to be determined by the six equations y 2 = j8-(32). Corresponding to each solution of this equation system we obtain a system of orthogonal surfaces, and the different systems thus obtained have the property of having their normals parallel at corresponding points. If a = ai + (21) + y (.31), b = /?, + y (32) 4- a (12), c = y a + a(13)+/8(23), (135.1) then c 1 = aX, z% = bfi, z.^ cv, and a, = 6(21), 6 3 = c (32), c 1 = a(13), 3 = r(31), 6 1 = a(12), C 2 --= fc (23), so that the ground form for the Euclidean space is ds 2 = a 2 rfu 2 -I- 6 2 cZi; 2 + c*dw\ (135.2) 136. An alternative method indicated. The functions c(, 6, c of ^, v, w must satisfy certain conditions which can at once be obtained by expressing the rotation functions in terms of a, b, c and their derivatives and using the conditions which the rotation functions "must satisfy. But we can AN ALTERNATIVE METHOD INDICATED 185 more rapidly obtain these conditions by just saying that the space defined by d 8 z - a *d u * + I>*dv 2 + c*dw* (136.1) is flat, and therefore (rkih) = 0. (130 . 2) The conditions then arc seen to be , 7 PI 7 <>> " These six equations if we could solve them would equally lead to orthogonal surfaces, and this is the usual method by which the problem of orthogonal surfaces is approached. There seems, however, to be an advantage in making the whole theory depend on one equation of the third order as we have done. 137. Three additional conditions which may be satisfied. We now wish to consider a special class of orthogonal surfaces, and we begin by inquiring whether there are any rotation functions which, in addition to satisfying the nine necessary conditions which all rotation functions must satisfy, also satisfy the three additional conditions (23) 3 +(32) 3 + (21)(31) = 0, (31), + (13) 3 +(32)(12) = 0, (12). 2 + (21) 1 + (13)(23) = 0. (137.1) If we take (23)EEu; + (31) = ^ (32) =x-, (13)E and 2 u' = v + iv, 2 u' = w -f u, 2 iv f = n + v, and, /being a function of the parameters t</, v', ty', denote respectively by / x , / 2 , 186 ORTHOGONAL SURFACES we see that the twelve conditions which the rotation functions now have to satisfy are expressed by 2/4 = - & = -27;s, 7, 3 = -2 *3 = -2^ f 17! = - 2^, ^ = -2iya?. (137.2) Now it is easily seen that these equations are satisfied by taking (23) = 4V%, + av%. (32) = (12) = -|/K; + ^V 3p (21) = i (137.3) where F is a function satisfying the 'complete' system of equations K ;14 + 2 v/CK 4 V 42 = 0, V. M + 2 VV~V~V K = 0, ;K 4 = o, ^+2^1^7 = o. (137.4) \Ve thus see that such rotation functions exist. A particular solution of such a system of equations would be obtained by toeing F< = 0) ^ a+2 v%lK ll -=0; (137.5) and in this case (23) = (32), (31) = (13), (12) = (21). (137.6) It may be shown that this solution corresponds to the particular solution of the original equation of the third order when we take z to be independent of iv. 138. Orthogonal systems from which others follow by direct operations. We must now consider the special property which the orthogonal surfaces will have which correspond to rotation functions satisfying the twelve conditions. We return to the original variables n, v, w in what follows. Let a, /3, y be any scalars which satisfy the equations 0,-=y (13), ia =a (21), y 2 = j8(32), (138.1) SPECIAL CLASSES OF ORTHOGONAL SURFACES 187 and let a, b, c be defined by o = a 1 +/8(21)+y(31), b = a + y (32) -f a (12), c = y a + a(13)+j8(23). We then have (138.2) Let a' = </! + 6(12) + c(13), 0' = 6 2 + c(23) + rt (21), / = c 3 + a(31)+&(32). We can at once verify that ^3=7' (13), ^ = ^(21), y' 2 = /8'(32); (138.3) and therefore c' = a'X + /S'/i-f y V will trace out another system of orthogonal surfaces. This second system is thus obtained from the first by direct operations not involving integration. We thus see that when we are given any one system of orthogonal surfaces of this particular class we can deduce by direct operations an infinite system of such surfaces. CHAPTER XI DIFFERENTIAL GEOMETRY IN ?i-WAY SPACE 139. Geodesies in w-way space. In order to see what kind of geometry we may associate with the ground form of an 7i-way space, we naturally think of the simple case when n was 2, and the space a Euclidean plane. The most elementary part of that geometry was that associated with straight lines ; that is, the shortest distances between two points. We are thus led to consider the theory of geodesies in our 7i-way space. We have . , f 2 T- = a ik -7 +a ik ~i --, j -r da ll * ds ds ds da ds ds d / dx- dx 1f . \ d = ds ( rt ds SX >< + a * -* **') - 8X >< ds For a path of critical length therefore we must have . , ds \ it d8 ds < d~ 8 ~ T^ ds 'ds Now ( 6) d Za ik dx p dx p// . and therefore dx: daw t GEODESICS IN M-WAY SPACE 189 It follows that 7? = and therefore a it ^ + (ikt) d p ^ = 0. (139.2) ' ds* as as Multiplying by a l P and summing we have X f + {ikp} _ -^ = 0. (139 . 3) We thus have TI equations wherewith to obtain the coordinates of any point on a geodesic in terms of the length 8. But the equations are differential equations of the second order ; and in general we can only solve them so as to obtain the coordinates in tbe form of infinite series. This is a practical difficulty and one of the reasons why we cannot have the same kind of knowledge of the theory of geodesies in Ti-way space that we have in Euclidean geometry of straight lines. The direction cosines of an element of length in ?i-way space are defined by dv ti> - ax p n _ i n / 139 4 \ i* ; * iJ 1 ,,/(/ i A o jj TI i 5 ds ' L v ; Going along a geodesic, therefore, we have and we see that, unlike the direction cosines of a straight line in a plane, associated with the form ds* = dx\ +dxl, these direction cosines vary as we pass along the geodesic. Thus we are familiar with the difficulty of keeping to the shortest course between two given points at sea, viz. a great circle. In this case the differential equations are soluble in finite terms; but even with* this advantage we should need 190 DIFFERENTIAL GEOMETRY IN M-WAY SPACE a continuously calculating machine to find the direction cosines at each point of the course. If the ocean instead of being spherical were ellipsoidal, we should not even have tho advantage of being given the equations of the geodesic in finite form, and the difficulty of keeping to the shortest course would be even greater. Now if we had built up our plane geometry by using the form d8* = dxl+x*dxl, the direction cosines of a straight line would also have varied from point to point of the straight line and yet we would not say that the direction of the straight line varied from point to point. The navigator on the ellipsoidal ocean might hope till he had learnt a little more geometry to mend the want of constancy in his direction cosines as the plane geometer could mend his by a proper choice of coordinates. He could not mend this want of constancy by any choice of coordinates, but though the direction cosines change in passing along a geodesic there is no need to think of the ' direction ' as changing. We will then say that the direction in an 7i-way space is the same all along a geodesic. 140. Geodesic polar coordinates and Euclidean coordi- nates at a point. We recall the fact ( 2) that any w-way space may be regarded as lying in a Euclidean ?*-fokl where r = %n(n+ 1), and that the vector z which lies in this r-fold, depending on the n parameters x l ...jc ny has the property that its extremity traces out our ?i-way space. In the 7i- way space, unless it happens to be merely a Euclidean space, we cannot think of a vector as lying in it : it is only tho extremity of the vector with which we are concerned. But at any particular point of the 9i-way space there is a Euclidean w-fold which we may usefully associate with the point. Let z be the vector to the point under consideration, and let z lt .*z n be its derivatives at the point with respect to x l ... x n , the parameters of the jfoint. TANGENTIAL EUCLIDEAN W-FOLD TO AN W-FOLD 191 Let 1 ... w be the direction cosines of any element of the n- way space at the point so that !** = 1, 040.1) then the vector defined by = 1 s l + . ..+%, (140,2) will lie in the Euclidean Ti-fold at the point. It will clearly be a unit vector sinco We shall call this Euclidean '}i-fold in which lies the tangential 7i-fold at the point. The coordinates of any point in the tangential ?i-fold ma}'' be taken as t . . . g n , where & = B ; , (HO. 4) s being a scalar. We establish a correspondence between the points of our ?i-way space and the points of the tangential w-fold by taking the coordinates of the 9* -way space to be a ... M . Consider the geodesic which starting at the point under consideration has the direction cosines l ... n . From the equation of a geodesic ff . --- . ds d* we see that the current coordinates are given by ' > (140 . 5) > where s is the arc from the initial point. Let z 9 be the vector which traces out the ?i-way space at the point x' l ...x' n> and let 9 denote the same vector expressed in terras of the coordinates & ... n - We have [see 4 for the notation] n = ^(*;-{^}r+'-o- It follows that 192 blFFERENTIAL GEOMETRY IN n-WAY SPACE and therefore the transformation formula is a>'n = <* ik -(kpi) l '8-(i\k)?8 + ..., (140.6) where + ... refers to higher powers of s. We thus have at the oriin that is, in the coordinates we have chosen the first derivative of each of the coefficients in the ground form vanishes. It follows that in this system of coordinates, which establishes a correspondence between the points of the tangential n-fold and the points of the ?i-way space, every three-index symbol of Christoffel vanishes at the point under consideration. As regards the four-index symbol (rkih)' we have = (rkih). (140.8) We may call this transformation a transformation to geodesic polar coordinates at a specified point. We can combine the transformation with any linear trans- formation in the tangential 71-fold. To do this suppose x 1 ... x n to be the original coordinates, taken to be zero at the point to be considered. Let i = ^vX* (HO. 9) where Cy.... denote constants. We can now so choose these coordinates as to make the coefficients take any assigned values at the point We can then apply the geodesic transformation, and can thus arrange that the coefficients a ik may have any values we like (pro- vided the determinant is not zero), and at the same time have all the three-index symbols vanishing at the point. In particular we caiiuso choose the constants that '<&='< (140.10) EUCLIDEAN OR GALILEAN COORDINATES AT A POINT 193 at the point and that the three-index symbols may vanish. Such a system of coordinates may be said to be the Euclidean coordinates of the n-way space at the point.* 141. Riemann's measure of curvature of 7i-way space. If we take the transformation where , 77, ... are fixed vectors at the point, we find that a' 22 = 'S*, (141.2) and (141.3) and therefore (1212)' (141.4) Now let us consider the expression on the left-hand side of this Aquation. In general the four-index symbol as applied to 7i-way space is not the same thing as when applied to the lower space in which the coordinates whose integers do not occur in the symbol are put equal to constants. But in geodesic polar coordinates at the point the equality holds, since ,the three- index symbols vanish. It follows (see 24, 37) that^ the expression on the left is the measure of curvature at the point of the two-way surface formed by keeping all the geodesic coordinates constant except two. The expression on the right is therefore the measure of * This system of coordinates has jjeen calld the system of Galilean cq- ordinates at the point. 2843 C 191 DIFFERENTIAL GEOMETRY IN n-WAY SPACE curvature of the geodesic surface that is, the surface formed by the geodesies through the assigned point, which touches at the point the Euclidean plane generated by the two vectors and rj. This is Riemann's measure of curvature of the 'ft- way space. We see how it is connected with Gauss's measure of curvature, and we should notice how in this respect the tangential 7i-fold takes the place of the mere tangent plane when n = 2. In the flat Ti-fold we consider all the Euclidean planes by taking 1 any two vectors in the Ti-fold. We see that these two-way surfaces have different curvatures and so different geometries. 142. Further study of curvature. The Gaussian measures for geodesic surfaces. Orientation. We have now obtained Riemann's measure of curvature and have seen how it is con- nected with Gauss's measure of curvature of a surface. We must now consider this curvature from another point of view. We saw that we were to consider the direction to bo the same at all points of a geodesic in u-way space. This leads us to define two 'parallel' displacements at neighbouring points x l .. f x n and x 1 + dx 1 ... # + dx n as displacements whose direction cosines 1 ... n and *-f d l ... n + d n are con- nected by the equations dP+ {ikp} gdx k = 0. (142 . 1) Thus in this sense of 'parallel' the tangents are parallel at all points on the same geodesic. It may be noted that the equation defining parallel dis- placements does not entitle us to say that If this equation system held, the tensor component P would be annihilated by every operator 1,2, ... TI, and therefore which could only*be true in flal space. FURTHER STUDY OF CURVATURE 195 Let be any vector in the tangential ?t-fold at x^ ... oc n and + d be tho 'parallel' vector in the tangential ?i-fold at n\ -f dx l . . . x n - We have df = {'(*.<* + [i& As we pass from the point of departure with an assigned value for , and the vector is carried parallel to itself, its value at any other point is defined by the integral 3. ik Jx k , (142.2) and this value depends on the path of integration. Consider the small parallelogram in the ?i-way space whose edges are parallel to the vectors and 77, the lengths of the edges being respectively a and 6. Wo want to find the change in f by integrating round the parallelogram. We have i = = '- where SjCp is the increment in the coordinate, neglecting powers of small quantities of the second order, and Z-tfc = (3-ifc)o + 0-tt/*+ (*^/ 3/fc+ (Ml S-ttL&V and therefore We thus have *'* = ^'*-)o+ [r (*-^+ {i>j - On the first edge at a point distant s this is K f *-tt)o+[^ (*+ {*/*'} -.)n on the second edge it becomes tf f s-)o+[^ (- +[r(^^ The change in ^ bj' integrating 196 DIFFERENTIAL GEOMETRY IN tt-WAY SPACE along the two edges is ) W]o-2 ( 142 3 ) If we had integrated in the opposite sense along the other two edges we should have interchanged and 77, a and 6, and we thus see that the change in by going round the parallelo- gram in the same sense is K'(-tt M -*-< M *)*V]o&> that is, [f f {#Ms,V] &. The change in ^ is therefore If is the angle between and (142.5) Let us now consider how the angle d is changed, if, keeping 77 fixed, we carry parallel to itself round the parallelogram. ab. It follows that <5d divided by the area of the small parallelogram is equal to ~ 6) That is, 5^ 'divided by the area of the small parallelogram is equal to the curvature of the geodesic surface which touches the Euclidean plane generated by the two vectors formed by the sides of the parallelogram. From the equation tff=f'*-ttk* we see that the rate of Change of f in parallel displacement FURTHER STUDY OF CURVATURE 197 in the direction of the vector is ^C Zt il^ We ma J express this result in the notation Thus we have . = fe *' It follows that and therefore (142.7) Kieumnn's measure of curvature may therefore be written ^-fcf* .^.. ____. (142.8) ||25""l?2l Again the rate of change of in displacement along the vector is just f 1 ^ 1 + ... + n <z> and therefore the vector itself may be written 0.. Here we may notice that the vector z unlike the displacement vector is not a vector in the 7i-way space but only in the containing ?*-way flat space ; the vector on the other hand marks a direction, or displace- ment, in the n-way space, although it has only an elemental length in this space. There is then as regards the vector z in its parallel displace- ment just the ordinary Euclidean ide'a of translation. 198 DIFFERENTIAL GEOMETRY IN ,71-WAY SPACE Riemann's measure of curvature may therefore be written -^- ^ (142.9) This is Gauss's measure of curvature -of the geodesic surface, made up of the singly infinite system of geodesic curves drawn through an assigned point at which we require the measure of curvature: the curves at the point all touch the Euclidean plane generated by the vectors ^ and 77. Riemann's measure of curvature has an ' orientation ' given by the vectors ^ and 77 ; and at the assigned point, by varying this plane, we get the different Gaussian measures. 143. A notation for oriented area. So far we have in using vectors only considered their products as scalar products. There is another product which we ought now to consider. When 7i = 3 and the ground form is that appertaining to Euclidean space, we know what the vector product means and how useful it proved in Geometry, but it does not seem to be capable of useful extension. We shall now think of the product of two vectors ^ and 77 as defining an area in the Euclidean plane formed by ^ and 77. This area has then an orientation, and we shall understand by 77 the area of the parallelogram whose edges are the and 77 drawn through the point. The angle the vector makes with the vector 77 being 0, by parallel displacement of the vector round the parallelogram whose edges are in the directions and 77, and whose lengths are ds and Ss, this angle is increased by z tt~'w~~ z t'<i z i]t which may be written ^ /* _ /y /? z tt z w z ft"nt dzSz, (143. 1) ' z,z t z z where dz, Sz represent* the sides ol the small parallelogram in magnitude and direction*. * ORIENTED AREA 199 We should notice that area has a sign as well as a magni- tude : we express this by the equation 17 + 1^ = 0. (143.2) 144. A system of geodesies normal to one surface are normal to a system of surfaces. If the direction cosines of a geodesic are l ,... w > we have seen (139.5) that the equations of a geodesic are The geometrical interpretation of these equations is that the tangent remains ' parallel ' to itself as we move along the geodesic. We can put the equations in another form, and therefore since -r as Now -(t and tberefore the equations of a geodesic may be written We now wish to consider the expression T , = ** Wo know, from the theory of differential equations, that the necessary and sufficient condition that T x dx^ may be rendered a perfect differential by multiplying by a factor is that ; 2 *" T ) should vanish identically for* all values of A, /*, v ; and that 200 DIFFERENTIAL GEOMETRY IN n-WAY SPACE the necessary and sufficient condition that T^dx^ may vanish, wherever (f>(x 1 ... x n ) = 0, is that 2\ A should vanish for all values of X, /z, v wherever Geometrically interpreted these are the conditions that the curves d ,j n ^==^==.,.= ^ (144.3) should (1) be cut orthogonally by a system of surfaces, (2) be cut orthogonally by the definite surface (,... ) = <>. (H4.4) We are now going to prove that if the curves are geodesies, and if the condition (2) is satisfied, (1) is satisfied also. \ rp *^rn Let -.Jsta, (144.5) and T p (97-) + T q (rp) + T T ( m ) = [p, q, r]. (144.6) Since A JL = -L. .* *? * ds ^x~ ~~~ Zx ds dx <>jc' cts v XQ (jts * u x^.. cts Now ~ GEODESICS NORMAL TO SYSTEMS OF SURFACES 201 It follows that Now and therefore \ve see that the first term in this expression vanishes, since A and /z are interchangeable. We therefore have It follows that d A. A. [7- A. r]) + ||- ((rj>) r x + [r, A, * > D d 202 DIFFERENTIAL GEOMETRY IN tt-WAY SPACE Since a \^i^ *> we have *r A = 1, and therefore T K ^- + x ~ - = 0. It follows that d r T Suppose now that over a given surface [p, q, r] = 0. We then have d r -, fc[P>1.r] = (qr) (p\) + (rp) (q X) + (pq) (r A). Since [p, q, 7 1 ] is zero for all values of the integers over the given surface, we have =0, 0, (sr) 9 (7*)> 0, and therefore (rs), 0, M> (F") that is, (pq) (rs) + (^r) (/?) 4- (rp) (g) = 0. (144.10) It follows that if [p, g, r] is zero over a surface it is zero everywhere, arid therefore if a system of geodesies are normal to any one surface they f are noimal to a system of surfaces. (P9) GEODESICS NORMAL TO SYSTEMS OF SURFACES 203 The direction cosines of the system of geodesies therefore satisfy the equation system It follows that K^aPi^^-aP^ = a qr gi r = 1, (144.11) and therefore "nttf = P - ' ( 144 12 ) 145. The determination of surfaces orthogonal to geodesies and of geodesies orthogonal to surfaces. We can now find the equation which (/> must satisfy when the surfaces = constant are those which cut the geodesies orthogonally. We have ^> = u P f ~-^=- (145.1) S V ' The equations of a geodesic being (131) . 5) d we must have that is, aV --A~ a** -= = 0, and therefore P f fc( ,a tx , A ) = 0, or =0. (145.2) Expanding, A (0, <^) = ___ A(^, A 2 A () that is, -r ** q 204 DIFFERENTIAL GEOMETRY IN n-WAY SPACE It follows that and therefore A (0) must be a function of $. Without loss of generality we may therefore say that A(0) = l. (145.4) We thus see that if we take any surface and consider the geodesies drawn from every point on it perpendicular to the surface, they are cut orthogonally by a system of surfaces (f) = constant, where A (0) = 1. In ordinary Euclidean space this is the theorem that the lines normal to any surface are cut orthogonally by the surfaces (f> = constant, where Conversely, let be any integral of the equation A (</>) = 1 ; then we shall show that the orthogonal trajectories of the surfaces (f) ~ constant will be a system of geodesies. It will be convenient now to think of an (>i-h l)-way space and to take, instead of the variables x l ... tf n +i> a new system of variables y l ... y n+l , where y + iS0, (H5.5) so that A(2/ n+1 ) = l, and to choose y l ...y n as n independent integrals of the equation A (y n+l , y) = 0. (145.6) Let the ground form of the (n -f l)-way space be 1>K<lynlyit* 1 = 1, 2,... n+l. (145.7) Since A (y ll + l ) = 1 and A (y n+l , y r ) = if r ^ (n+ 1), we see that 6 n+1| n+1 = 1, & n+1| r = 0, and therefore the ground form is rfVi + 6 <A<fyfc> 1- = l> 2 > - ^ ( 145 8 ) It will therefore be convenient to take as the ground form in the (TI-J- Ij-way'fcpa^e du*+btydXidx kt i = 1 ... 7i, (145 . 9) where 6^ is a function of x l ... cc w and u. GEODESICS NORMAL TO SYSTEMS OF SURFACES 205 The surface ih = is any arbitrary surface in the (n + 1)- way space, and when u = we may write b ik = a^ . We may consider a^dx^lxj, (145.10) to be the ground form of the u-way space deduced from the (71 + l)-way space by putting u = 0. The lower ground form may be said to be the ground form of a surface in the higher space. By varying u we obtain a series of surfaces cut orthogonally by the curves whose direction cosines are given by l = 0, 2 = 0,...^ = 0. These curves are geodesies, since {n+1, ti+1, p] = 0, p = 1 ...n. It will be noticed that the first of the surfaces cut orthogonally may be any whatever, but the other surfaces are given by A(tt) = 1. (145. 11) When we know the geodesies normal to u = 0, we know the whole series of surfaces which are cut orthogonally, or at least can find them by quadrature, since '' = < (145.12) We obtain the geodesies, on the other hand, by the solution of the linear equation A (u, v) = 0, when we know u.* 146. A useful reference in (n + l)-way space. We have shown (145.9) that the ground form of any (/i+l)-way space may be taken to be tin* + bftdxjdxjg , * = 1 . . . n, where b^ is a function of x l ... x n and u. The surface u = is any surface whatever in* the (u+1)- way space. By drawing the geodesies perpendicular to this surface we obtain a series of curves which are cut orthogonally by the surfaces u = constant, u being the geodesic distance of any point from the surface u = 0. * [At this point hi the author's MS. Iliere ia a memorandum ' New Chapter '.] 206 DIFFERENTIAL GEOMETRY IN 71-WAY SPACE The surfaces u = constant are said to be parallel surfaces, and we have A (u) = 1. Travelling along any of the geodesies from the surface u = 0, only u varies. It will, however, be found useful to consider a more general system of surfaces in the (n+ l)-way space. We therefore consider any system of surfaces whatever in this space, u = constant, where we no longer have A (u) = 1, and by taking the orthogonal trajectories of these surfaces, as the parametric lines x l = constant, ... x n = constant we may take the ground form of the space to be 2 <Zi6 2 + b ik dx { dx k , i = 1 . . . n. (146.1) The orthogonal trajectories are now no longer geodesies. The function b ik depends on the coordinates x lt .. oc n arid u, and, when u = 0, b ik = a^.. We now wish to consider the two round foi'ms dxid^ (146.1) and ^i k dx { d^ k (146.2) in connexion with Christoffel's symbols, where after calculating their values for the higher space we put u 0. We can obtain the special case of parallel surfaces by putting 0=1. We shall thus be shown how to generate the (n+ l)-way space which as it were surrounds any given 7i-way space. When we place the suffix b outside a symbol this will indicate that the symbol belongs to the higher space. The suffixes will always be l...n. When we have to consider the suffix which should correspond to the variable u it will be denoted by a dot. Let * ^ = _ 2 /2< fe f (146.3) We see that (ikh) b = (Usk) a ; (ik -) b = fl ilt tj>; (i- k) b = - fl ik $ {ikh} b = {ikh} a ; (rkhi),, = A USEFUL REFERENCE IN (n 4- l)-WAY SPACE 207 in tensor notation, 147. Geometry of the functions /2^. If we are given any n-way space we shall see that it may be surrounded by what is called an (71+!)- way Einstein space. We shall define this space shortly. A particular n-way space may be surrounded by a Euclidean space of n + 1 dimensions ; but before wo consider particular kinds of this (714- l)-way space we had better consider the geometrical meaning of the functions /2 t ^ which together with (f> are to generate the space. With this end in view let us consider two geodesies going out from the same point x l ... oc n , u = 0, and having the same direction cosines *... n , at this point, the first geodesic being in the 7i-way space denoted by a, and the second in the (n+ l)-way space denoted by 6. We have (140.5) for the current coordinates on these geodesies x\ ...x' n and x'\ ... x" n and therefore, neglecting terms of the third order in the arc s, we see that the coordinates are the same for the two geodesies. But for the first geodesic the coordinate u is zero, and for the second geodesic The distance between two points, one on each of the geodesies, is therefore 2 ' 208 DIFFEBENTIAL GEOMETRY IN n-WAY SPACE 8 being the distance of each point measured along its own geodesic from the initial point. This mutual distance we may consider to be normal to the geodesies if we neglect terms of the third order in the arc. The curvature of the first geodesic is defined by Voss as the ratio of twice this distance to s 2 . This is obviously a proper definition, agreeing with the ordinary definition when we are dealing with Euclidean space. We therefore have ^L\^ax x ax^ /1 . O x m - llTt* . l ^f/'^A^A* The curvature of the first geodesies may be called the normal curvature in the (tt,+ l)-way space of the surface u= in the direction dx lt dx 2 , ...dx n . Looked at in this way we may write our formula 1 = A/jx^ t (147.3) jR ^Au^A M To get what we may call the directions of principal curvature we require the directions dx t which make ^ critical. The directions of principal curvature are therefore given by (a x/i -H/2 A/l )cte /t = O f (H7.4) where the values of R, the principal radii of curvature, are given by the determinant |a A/4 -JB/2 AM | = 0. (147.5) We shall.now show that the directions of principal curvature are in general mutually orthogonal. At any given point we can choose the coordinates so that corresponding to the principal radius R^ only the coordinate X} varies at the given point. We therefore have at the point ' a ilt = Ri.fl'j.. IK K IK' GEOMETRY OF THE FUNCTIONS 209 If, then, all the radii of curvature arc distinct, a ik = n ik = 0, if i&k, (147.6) and therefore the coordinates are mutually orthogonal at the given point: that is, the lines of curvature are in general mutually orthogonal. 148. The sum of the products of two principal curvatures at a point. We now wish to obtain an extension of the well-known formula of Gauss 1 __ (1212) for a surface lying in ordinary Euclidean space. Consider any determinant a a nl nn and the corresponding determinant If \a\ denotes the first determinant we see that any of its minors is equal to | a multiplied by the complementary minor of the second determinant. Expanding the determinant (147 . 5), or say we see that the determinant divided by \a\ is equal to x x 2 '* .*/ i . '* / v! Ill' t'li i'7 / / / / / \ a (a ? *a / ' A -a ?/ 'rt f ^(tt H a ?fk -a rh a i k )-..., (148. I) the numerical factors T-TT,- > TT2 "TV*" being introduced (1!) (2!) (3!) in accordance with the convention about repeated factors. We therefore have for the principal curvatures (148.2) ^-a,;AJ. (148.3) wit * K e 210 DIFFERENTIAL GEOMETRY IN Vi-WAY SPACE Now consider the expression, a tensor component clearly, and similarly B ri == b kh (rlchi). We have seen in 146 that From the last equation, we see that Vy /2 rA /V (148 . 4) ., From the second equation, we see that (B,.) b = a'<>><l> (n fh . k -n hk . r ). (148 . 5) From the first and last equation, that ^. (H8.6) The expression a ri A r ^ is an invariant which we denote by A. We thus obtain or, interchanging i and h, E = A + a'*a*' (/2, A /2 i7 ,-/2 and therefore 2J5 = 2A+(a r *a 1sh -a rh u ki ) (Sl rh Sl ik - (148.7) It follows that 2B= 2A + 4S-~ L jy +40~ 2 5... (148.8) Kemembering that B is an invariant in the (n+ l)-way space, and B.. a tensor component in this space, and that A is an invariant in the ?i-way space obtained as a section of the (7i+l)-way space by the surface u = 0, we may express the result at which we have arrived in the following way. SUM OF PRODUCTS OF TWO PRINCIPAL CURVATURES 211 Consider any (>i-h l)-way space, and a section of it by any surface. Let g l ... n ~ Hl be the direction cosines of this surface, re- garded as a locus in the (n+ l)-way space. The diiection cosines are connected by the identical equation 1 = !>,#? The expression V ljt (rkld)^ r g l is an invariant of the (ft + l)-wuy and the surface we have chosen. When we take the ground form ^du^-i-b^dx^dx^ and the surface u = the expression becomes 0~ 2 ./?.., since * ... g n are zero, and 02 71+1 7* +1 __ i If B is the invariant of the (n+ l)-way space, and if A is the corresponding invariant of the /i-way space which is the section of the (71+!)- way space by the given surface, then what we have proved is the following. The sum of the products, two at a time, of the reciprocals of the principal radii of curvature of the surface, regarded as a locus in the (>i-f l)-way space, is equal to -I (B-A)-b l(h (rkhi) b g r tf. (148 . 9) 149. Einstein space. Suppose, now, that instead of taking any surface we choose a surface whose direction cosines satisfy the equation %J3=b Ut (rkhi)g r g ! ', (149.1) we shall have 1 i^ + J 7r/ ^=0. (149.2) ^i^k For the case n = 2, this becomes Gauss's well-known formula t (1212) li l ^ 2 a n a 22 a j 2 If, then, the (?i-f l)-way space is to be such that for all surfaces lying in it the formula of Gauss will hold, the equation i^ = **?' must be identical with 1 = 212 DIFFERENTIAL GEOMETRY IN n-WAY SPACE We must therefore have B H = Wb H , (149.3) and in consequence of this that is, B = %(n+l)B. (149.4) Leaving aside the case when n = 1, we must have 5=0, (149.5) and therefore B ri = 0. (149.6) A space with this property is what is called an Ein.stein space. It is interesting to see how from mere considerations of purely geometrical ideas we should be led to Einstein space. 150. An (H+ l)-way Einstein space surrounds any given 91-way space ( 150-4). We shall now show how, being given the ground form of any ?i-way space, we may obtain the ground form of a surrounding (n+ l)-way Einstein space. We look on a ik . t . as functions of x l ...x n and u whose values are known when it = 0. We have if possible to determine functions /2 2 - fe ... and <p which will satisfy the equations a lh (fl tll . h ~fl hl .. t ) = 0, (150 . 1) (150.2) 6 = -2/2^0, (150.4) when u ~ 0. If we can find such functions, we take and thus find the Einstein form 2 cZu 2 + bftdxidxk (1 50 . 5) in the immediate neighbourhood of the surface u = 0, that is, in the immediate neighbourhood of the given n-way space, DETERMINATION OF SURROUNDING EINSTEIN SPACE 213 and proceeding thus by the method of infinitesimal stages we ultimately obtain the Einstein space which we require. Let A^ = a xt {tppn}. (150.6) We shall first prove the fundamental identity Employing the geodesic coordinates at a given point, and therefore in this system of coordinates we have at the given point (150 ' 8) \a >A Similarly wo have ft A and therefore A.. = 0. (150.7) The required identity holds therefore universally, since it is a tensor equation which vanishes for geodesic coordinates. 151. We now transform the functions f2 ijt ..., which are to be found thus. * 214 DIFFERENTIAL GEOMETRY IN n-WAY SPACE Let V* = a kt n it , and therefore fl il{ = ft ki = a^V* = a a F x , Wilting F for F we now have x=V, (151.2) that is, multiplying by a A ^ and summing, #(^+ The third equation is and by aid of the second equation this may be replaced by A = FF-F 2 . (151.4) We thus have the equations V* V A 1/7* y x V2 K A*A %> ^ K A ^A 1 "" K > and, writing <fy for a x *0 ^ in the other equation (151 . 3), Tl:ese are only assumed to hold when u = 0. 152. Noticing that = A 2 (0), we have $. x = Aa x< 0. Mf = a xt \p.<f> t and therefore ^-A+'^^A ^^^2(0)- ( 152 - DETERMINATION OF SURROUNDING EINSTEIN SPACE 215 We now operate with A on the equation where A = + {t\8}(()-{p\t} (<), Ox \ s being the upper integer here A itself so that we employ another integer 8 and p. the lower integer in V^. We easily verify that A-~ = Ax-(')jLj$x8) +().- <ht <)', w du ' J V c)u and therefore .,. (152.2) Now, since (4 . 3) x/1 a^ s = ej-, we see that . - a^ =2<j> a" Vf, (152.3) V Li and ~{rkh] = ~a we see that *. (152.4) Since . (152.5) 216 DIFFERENTIAL GEOMETRY IN n-WAY SPACE Now if we multiply each side of this equation by 7l and sum, remembering that we get Vl = v 1 ;. (t k v f + Ff . ,) - 0,. n vl - <f>, : V;. vi Fl'.,+ F!'.<) (152.6) We deduce from the equation for -- {rkh} that *~ { rklcl = <t >/ V' l . + <t>V l r . t -<t> l .V-< = -/>, F-0F,. - (152.7) Combining the formulae \ve have proved, we now see that *- v^ - v h , (<t> vl ., + ^ v I) - 7; ( We also have - V = </> (A + V 2 ) -f A 2 (0), o!6 and therefore A F a = M (4 + V' 2 ) so that r- - (1 A -^-2FF^. (152.9) Now and therefore ^ M + 2 FF^, = 2 7* V'l . M . We have proved (150.7) that 4* . K = %A^ and we have K. A ='F /J ; DETERMINATION OF SURROUNDING EINSTEIN SPACE 217 it follows that we have ^(V^-VJ = 0. (152.10) 153. Now A is an invariant for any transformation of the coordinates x 1 ...x n in the ^-way space, and so therefore is j-~. We shall therefore, in finding an expression for --> employ geodesic coordinates at a given point and thus materially simplify the necessary algebra. and therefore and we proved in the last article (152 . 7) that Hence, in geodesic coordinates, - - is equal to ' Now 218 DIFFERENTIAL GEOMETRY IN n-\VAY SPACE and therefore Xt) V 1 ,, %0 (153.2) It follows that - is equal to , (F) + 24'/ F,>. (153 . 3) Again, a^r^'T*, ' and therefore ^^ n-x = 0/a^ T^., = a">, F^ = A (0, TO, ^^ so that -r is equal to %u ^ -2FA 2 (0) + 2^r^ + 2^F> (153.4) Since, then, the equation ^F^ (153.5) is expressed in invariant form, it is true not merely at the point, whose geodesic coordinates we employed, but universally. Now so that (F^F^-F 2 -vl)= 0. (153.6) SUMMARY 219 154. If, then, we are given any 7i-way space, and we find functions V\ such that which will satisfy tho equations ^x=^, A=VlV*-V, (154.2) and if, taking arbitrarily any function <p of # 15 J\ 2 , ... x n and a new variable u t we allow a^. and Vl to grow in accordance with the laws ^ = -20^ (154.3) ^ = <j>(A i l + VV'' i ) + a'<j>. H , (154.4) <H(/ taking as their initial values when n the given values ot a ik in terms of ^ ...#, and the values initially found for Vf, the equations V ^.^=V^ A = V*V*-V* (154 . 2) will remain true when u takes any value \vhatever, and the form 2 c?u 2 + a {u dx { dx u (154.5) will be the ground form of an (u-f l)-way Einstein space. The surfaces u = constant may be any whatever in the Einstein space; and we see (149 . 2) that the property of this space is that the sum of the products two at a time of the reciprocals of the principal radii of curvature of any surface in this space is equal to \ A , where A refers to the n- way space given as the section by the surface of the (ti + l)-way Einstein space. CHAPTER XII THE GENERATION OF AN (w + l)-WAY STATIONARY EINSTEIN SPACE FROM AN oi-WAY SPACE 155. Conditions that the (>i+l)-way Einstein space surrounding a given )i-way space be stationary. Wo have shown tbat any rt-way space is surrounded by an (n-f-l)-way Einstein space, and that the equations which lead to the Einstein space are V K V' A V x V^ V 2 Y ' > " ' where A* = a xt {t The Einstein space has the ground form <t>' 2 du 2 + bji.dxidxj., where bjj s is equal to a^ when u = 0. We now inquire what properties the n-way space ground form must have if (f) and by c are to be independent of u. Clearly the necessary and sufficient conditions are that V\ = 0. (155.1) Wo therefore have -4 = 0, <J)A^ + 0^ = ; that is, 0{X/}+0- x ^=0; A 2 (0) = 0. (155.2) We now 'want to transform the ground form a ik dx i dx k and the function (f> by the transformation <*=&*". <^= (2 - n)r - 055.3) We have a"- = e~ zv l ik , THE APPROPRIATE W-WAY FORM 221 Let* 0'* where QJ = and therefore 6- a -ft. We also have A (0) = 0% A 2 (0) = 0'.,. It is easy to verify the following relations : 0r,./,-0,Vi = (^-2)d. th + a rh \(0). (155.4) Now W}a={^'h+^, and {rkih} = ^- {rifc} - .-- { r/Jj} + {r} [thk] - |r/^] {fit}, d^ A MI and therefore We also sec that ) a -(n-2) (A and therefore (A 2 (F)-(ft-2)A(F)) rt -(A 2 (F)) 6 =0. (155.0) NOW F;,. /( -F,V,+F!.,F;,-F;,,F,, = (n-2)(F. rA +F r F 7i ) + a,. 7( (A 2 (F)-Oi-2)AF), and therefore {rtth} a = [rtth} b + (n-2)(V. rh +V r V k ). (155.7) It follows that, since -1) V r V h = 0. (155.8) * [This is not the introduction of some new function 0, but an assign- ment of meanings to 0^. and 0* in connexion with any known 0^. The meaning of f^ is that assigned in ^.] * 222 AN (>l+l)-WAY STATIONARY EINSTEIN SPACE For the form b^dx^dx^ we therefore have A 2 (F) = 0, {rtth} b + (n-l)(n-2)V r V h = Q. (155.9) If we can find a ground form to satisfy these conditions, then <* = & Sr . = e( 2 -"> r (155.10) will give a ground form which will lead to an (7i + l)-way Einstein form, the coefficients of which will be independent of u. 158. Infinitesimal generation of the (?i-fl)-way from the n- way form. In order to simplify the problem of finding the ground form b^dx^lx^ we shall regard it as an (n+ 1)- way form and bring it to the form * <p*du~ + bfadzidXk , J = 1 . . . n, (156.1) as we have done before. The equations A, (V) = ; {rtth} + it (n- 1) V r V h = now become, if we take V Vn ( 1 n) u, {rtt-} = 0. (156. 2) If we regard the form as generated from ^ i^dx^Lr^ wo have ( 146) the equations We therefore have ., i - ^n^-a^a^n^n^, (ise . 3) = a"'(n r!t . k -n Mi . l ), (156.4) o = 4>(A H + u Ki'(n ri n hk -n rK n i d- * The tf used here is not the ^ of the Einstein form. GENERATION OF THE (tt-f l)-WAY FORM 223 The first equation may be replaced by - 0- 2 = A + a^ (fl fi fl^ - fl,^ a* 1 . The geometrical meaning of this equation, since it may be written is 12 that is, the sum of the products two at a time of the principal curvatures of the equipotential surface V= regarded as a locus in the (n+ l)-way space is equal to %(A + 0~ 2 ). Making the transformation to the functions VI we have ^=-20a f . v y?, (156.7) and, from [A 2 (u)] b = 0, we have that is, - X/1 /2 A A-' -0- 4 ^ = 0, 16 or - F0- 1 - </>-> ^ = 0; c)U so that ^ + F0 2 = 0. (156.8) We have, a denoting the determinant of the form a ili dx /i dx k , t>ct j*ba - - = aa l - - - en du = - 2 aa i/f f2 ih (f) = -2a V<f> ; and therefore ^ (a0~ 2 ) = -2aF0- 1 -2a0~ 3 ^; o '\jj v 16 so that the equation -~ + F0 2 = may be replaced by (<^0~ 2 ) =- 0. (156. 9) 224 AN (>l + l)-WAY STATIONARY EINSTEIN SPACE We have the equations As earlier, we therefore see that (156.10) and this ia equal to zero, since 0- 3 <^ = i^-J 7 X>+ FF x- We also have, from what we proved earlier (153 . 5), and we have i. (0-2 + F 2 - 7$; F^) = - 2 tfr 3 1| + 2 F (0 (vl + F 2 ) + A. and therefore (^ + 0~ 2 + F 2 - F ) = 0. (156.11) We thus see that the required (7i,+ l)-way form can be generated from any w-way form innnitesimally by choosing Fl and to satisfy the equations F* =F,,; A + 4>-*+V*-V*V$ = Q. (156.12) RESTATEMENT AND INTERPRETATION OF RESULTS 225 157. Kestatement and interpretation of results. We may now restate the result at which we have arrived. Let a^dx^dx^ be the ground form of any 7i-way space whatever. Find functions Ff such that a^ K V i = a iK V k , and which satisfy the equations F*. x = T^. Define a function by the equation where A=a Let the coefficients a^ and the functions and Ff grow, with respect to a new variable u, according to the laws having as initial values, when u = 0, the given values of a^ c in terms of x^ ... a; M and the values initially found for F* and (/>. The equations I/A __ rr H y - y will remain true when u takes any value whatever ; and, a denoting the determinant of t^..., a</>~ 2 will remain a function of x l ... x n only. Tho (n+ l)-way form (/> 2 du 2 + a^d.^dxj., i = 1 . . . ?i, will now have the properties A 2 (u) = 0; {.ft-} = l; {rA}=0; {r^-*} = 0. Transform now to any new variables which we may still denote by x l ... x n , % n+l , and let n (n 1) = and thus let the (71+ l)-way form be 226 AN (^+l)-WAY STATIONARY EINSTEIN SPACE It will now have the properties where A 2 (7) = 0. From this (n+ l)-way form let us pass to the form bftdxidxfr where b ik = a ih e 2v y and let 6 = ^ l ^ v . We now have, for the (u-f l)-way form Iwlxtdxk, [ = 1 ...71+1, or = 0, where ^ = ft*' {rtf/ij, = 6^ (9.,^. The (?n-2)-way form Q^du^-^b^dx^lx^ will now bo an Einstein form and the coefficients b^ and 6 will he inde- pendent of 16. 158. A particular ease examined when n = 2. As a particular case we might consider what properties the n- way space must have if in the (u + l)-\vay form which it generates, namely 02</ w 2 + a^.d^d^. , (158.1) the coefficients a^. and the function are to be independent of u. We must have, as the necessary and sufficient conditions, A -4 0" 2 - 0, 0^ + ^'0. //t =0, that is, A H- 0~ 2 0, 0(A^}+0. ;At = 0. (158.2) Now the chief interest of an Einstein space is when its dimension is 4. We shall therefore only consider this special case when n = 2. We thus have that is, '20 2 A r r= 1. (158.3) A PARTICULAR CASE EXAMINED WHEN U = 2 227 The other equations become ^a ll A"-0. 1l = 0; 0a 12 /t r -0. ]2 =0; 22 tf-0. 22 = 0. * (158.4) We wish to find the properties of the two-way form which will satisfy these tensor equations. The element of length on the corresponding surface* we take to be given by ds * = 2e - e dudv. We then have {111} + ^ = 0, {222}+0 2 = 0, {112} = [121} = {122} = {221} = 0, and A" = e*0 12 . The equations which have to be satisfied will now be 0,i + 0i0i = 0; 200 I2 = C -'; 23 +0 a 2 =0; 20V0 ]2 = 1. (158.5) We should notice that the suflixes in these differential equations denote ordinary differentiation, and not tensor derivation which would be indicated by the dot before the suffixes. By means of the equation 200 12 = e~~ wo can eliminate from the other three equations, and we see that they reduce to the two equations 00i0ii2 = 0i2(00ii-0l)> 0020221 = 012 (0022-01)- (158.6) These two equations may be written ^- ( lo 0i2 + l<>g -log 0,) = 0. Consequently *' 20 and ^ 20 (158.7) 01 02 are respectively functions of v only and of it only. Wo do not IOFO in generality by assuming that 0J20 = 01 = 02 and therefore = F (u + v)\ where F"J? = P . 228 AN (tt + l)-WAY STATIONAKY EINSTEIN SPACE The ground form of the surface is therefore <fc a = iFdudv, (158.8) where F is a function of u + v given by F" F = F'. If we take the parameters on the surface then ck 2 = F' (x) (dx 2 + dy*), (158.9) and F" (x) F(x) = F' (x). (158 . 10) The surface is thus a particular case of a Liouville surface. 159. General procedure in looking for a four-way stationary Einstein space. In general, when we want a four- way Einstein space of the form 6 2 du 2 + b ik dx i dx k , j. = 1, 2, 3, in which the coefficients b ik and are to be independent of u, that is, what is called the ' stationary ' form, we begin with the ground form ail dx\ + ^ a^dx^x^ a^dx\ . (159 . 1) We then find in any way three functions /2 11S /2 12 =r /2 21 , /2 22 of the parameters x } and # 2 which satisfy the tensor equations /2 n . 8 = /2 ls . lf /2 2ri = /2 12 . 2 . (159.2) We define a function by the equation /2 n /2 22 + /2? 3 . (159.3) We now let the coefficients a ik and the functions fi ij grow with respect to a new variable u in accordance with the laws 7J? = - 2 * fl tt' < 159 - 4 ) 75?= *' ifc - 15 The equation which defines <f> will be unaltered, and the equations /2 U . 2 = B^^, ^^ = B M . g (159. 2) will remain true. * r QUEST OF FOUR-WAY STATIONARY EINSTEIN SPACE 229 We thus attain the three-way form 4> 2 cZu 2 + a n dx\ -{2a^ 2 dx l dx 2 + a^dx\, (1 59 . 6) in which in general </> and the coefficients a^ will be functions of #!, # 2 , and u. In the particular case we considered in the last article, when the two-way form appertained to a particular class of Liouville surface, the function </> and the coefficients a^ will not involve u. But in all cases the three-way space with the form will have the property A 2 (u) ~ 0, (159 . 7) {ittk} = 0, {itt-} = 0, {} = !, (159.8) and therefore (2323) = -fa 22 ; (31 31) - _ * ; (31 23) - Ja 12 ; (23 12) - ; (31 12) - ; (12 12) = " (159.9) From this three-way form we can deduce the ground form of a stationary four- way space by the rules we have given in the general case. We should notice that if we begin with the proper Liouville surface, the Einstein stationary form at which we arrive can be, by a proper choice of the parameters, thrown into a form in which all the coefficients will be functions of two parameters only. 160. Conclusions as to curvature. The three-way space with the form is such that, if we regard the surface u = constant as lying in it, the product of the reciprocals of what we have called its principal radii of curvature is J/11 UoO """"" U T 9 11 22 - f 1 - (160,]) a ii a 22~~ a f2 * We must not confuse thgse racHi of curvature with the 230 AN (n+l)-WAY STATIONARY EINSTEIN SPACE radii of curvature of the surface n = constant regarded (as it may be) as lying in Euclidean space. We have in fact the theorem : * the product of the reciprocals of the principal radii of curvature of the surface, u constant, is equal to ]{ --J-0-*, (160 . 2) \vhere K is Gauss's measure of curvature/ Riemann's measure of curvature corresponding to the vectors and 77 which lie in the tangential Euclidean space at any point is ( 141) {]GO 3) i , l i i )' where the direction cosines of the vectors and 77 are re- spectively \ 2 , 3 , and 7/ 1 , r; 2 , rj''. If the vcctois and TJ touch the surface u = constant, this becomes -|0~ 2 . If their plane contains the normal to the surface it becomes \ " 2 . In the particular case when we start with the proper Liouville surface these are respectively K and K, where K is Gauss's measure of curvature. CHAPTER XIII it-WAY SPAOK OF CONSTANT CURVATURE 161. Ground form for a space of zero Rlemann curvature. We shall now consider the simplest form in which the ground form of a space may be expressed in which Riemann's measure of curvature is zero everywhere and for all orientations. For such a space {trpr/} (161 . 1) for all values of the integers. Consider the system of differential equations 0.^ = 0. (101.2) We have 0.^,. = 0, <f>. prq = 0, and therefore (* <] <]?') 0,, > that is, tlrfrq] fa = 0. (161 .3) A system of equations with the property that no equation of lower order can be deduced from them by the processes of algebra and of the differential calculus is said to be ' complete '. The necessary and sufficient conditions that the system of differential equations 0.^ = may be complete is then -\ptrq] = 0, (161 . 1) that is, that Riemann's measure of curvature is everywhere zero. If 'ii and v are any two integrals of the complete system -0, (161.4) and therefore, since A (?(, v) is an invariant, it is a mere constant. 232 ?i-WAY SPACE OF CONSTANT CURVATURE If, then, we take any n independent integrals of the equation system 0.pg = (161.2) as our new variables, the ground form will take such a form that each a^ is a mere constant. The ground form can therefore be so chosen as to have the Euclidean form d*l + ...+ds* m (161 .5) 162. Ground form for a space of constant curvature for all orientations. We next consider the ground form which corresponds to a space for which Rieinann's measure of curvature is the same constant for all orientations. We have (Xw) = K (a^a,, p -a^ V ), ami therefore [Xtpp} = Kfr^-a^fy. (162.1) If, then, t ^ p and t ^ (a, we Lave :X/z/jj=0, (162.2) and (A2W}=A'a AM " (162.3) it p is not equal to p,. Here the repeated integer p is not to have the usual implication of summation. Consider now the system of equations u ftq + Ka l)q u=0. (162.4) We see that tho system is complete : for u. tr + Ka )q u r = 0, r u q = 0, and therefore \ptrq] u t + K(a ljq u r a pr u^ = 0, and for a space such as we are considering this is a mere identity and not a differential equation of the first order. The system is therefore complete. Now let u be any integral of the complete system (162 . 4). We have ? & u = fa^ = - K ( CONSTANT CURVATURE FOR ALL ORIENTATIONS 233 and therefore Au -h Ku 2 (162.5) is a mere constant. As u does not satisfy any equation of the first order, being defined as any solution of the complete system, we can choose ib so that A (u) + Ku 2 is eipial to zero at the origin, and therefore zero everywhere. Let A r = - j (102. G) We now have A (log u) = />2 Take now as now variables where A (//,, 7/ 7 .) = 0, & = 2, ... u. The ground form will take the form dx'[ -f a^dx^dxj.y * 2, ... a, i r/ /"/'"' ami, since '^-^r/ "TJ-T" and u we have ! tk 1 \ H j~- = ; that is, (ikl) + - = 0, t> 2 or - (log ay.) -p It follows that a i1f = e" />,-/., (102 . 7) l/t t/l. 7 \ / where b^ is a function of .i' 2 ... x n only. As regards the form b.^dx-dxi., ' = 2, ... ?,, wo see that, since ' 2x i (rldh) a = e"^ (rkih) b + {ikl} (rhl) - {hkl } (ril) H L 234 U WAY SPACE OF CONSTANT CURVATURE and since (162.1) wo must have (rkih)^ = 0. (162 . 8) It follows that the ground form of a space of constant negative curvature may be taken as dxl+e (tteS + ...+<&,;). (162.9) By the substitution the form may be written 7) > (102. 10) The corresponding form for a space of constant positive curvature may be taken as 7?2 f -(-<Lx\+dx\+...+dxl). (162.11) Xi 163. Different forms for these spaces. We may find other forms for these spaces. Taking the case of positive curvature, instead of choosing u so that A(u)+ Ku* = 0, we may choose u so that (163.1) /> Let u = cos z , then A (x^ 1, and the ground form may be taken dx\ + a ik djCidx k , [ = 2, ... n. (163.2) Since u we now have (i/ol) + %^ cot ^J = 0, DIFFERENT FORMS FOR THE SPACES 235 and therefore a v . = sin 2 ^k- ; ., (163 3) 1 JLi where b^, depends on #' 2 ... x n only. We have (ikl} (rhl)-{hkl} (ril), and therefore rf^ = ;* ^ (rA tt )6 + cot* 5 " ' *"H^4^ ; that is, (,*;/o,. = *''**-> V The ground form may therefore be written where b^dx^dxj. is a ground form in o^...^ only, with the same constant positive measure of curvature. It at once follows that the ground form for a space of constant positive curvature may be written /-V> /Y* dx\ + sin 2 '! tU-r t + sin 14 -.! sin 2 '-J dx\ + . . . ; /i li JK or perhaps better as 7i 2 (c/a;? 4-siir a.y/.r? -|- sin-a. t 1 sin 2 a i 2 (/iC5 -f ...). (1G3 . 5) A form obviously equivalent would be li 2 (dx* + cos 2 a\dx^ + cos 2 a^cos 2 xjlx\ + ...) (163.6) The latter form when applied to a space of constant negative curvature would become -R* (dx\ + cos 2 ^^^ 2 + cos 2 a\ cos 2 X^IJL\ + ...), (163.7) and this may be written R*(dxl +cosh 2 .r l (te| + cosh 2 ^ cosh 2 x^dx\ + ...). (163 . 8) The surface ^ = constant, that is, the (n l)-way space x. z ... x n , regarded as a locus in the oi-wny space of constant curvature given by the form (ta| + ..., (1G3.9) 236 71- WAY SPACE OF CONSTANT CUKVATURE has all its principal radii of curvature equal to n x* H tan -73 9 M and any line on the surface is a line of principal curvature. 164. Geodesic geometry for a space of curvature -f-1. We shall now consider the geodesic geometry of a space whose curvature is positive unity : that is, the space corre- sponding to the form dt? = d-jci -f sin 2 aj,^? +sin 2 aJ 1 sin 2 a3 J rfo; + .... (164 . 1) We shall first find the equation which a path must satisfy if it is to he stationary with respect to variation of the coordinate x l . If we write for -4-* we must have * <is and therefore, since 1 = x\ -f- sin 2 x v x\ -f sin 2 #, sin 2 c. 2 ? t + > wo have ^ = cot^ (1 -x\). (1G4.2) It follows that - l . sin 2 ;r x (x\ - 1 ) = 0, and therefore cos j\ cos a x cos (s + l ) 1 (1G4 . 3) where a a and l are constants, and s is the arc measured from some point on the path. It follows that _- - - - r/ (1 COS'' Ofj COS^ ( -I- a ))" -, tan (s 4- f,) l ----- (164.4) T ^ A - Let **, = tan 1 sin then we have ds'i = rfe^ -f- sin 2 # 2 cZ#o + . . . . (164.5) Here 8 l is the arc in, an (/i-l)-way space of curvature positive unity, and if 8 is to be stationary for variation of # a , then 8 1 must also be stationary. ' GEODESIC GEOMETRY FOR A SPACE OF CURVATURE + 1 237 Proceeding thus we see that the equations which define a geodesic are CO s x l = cos a t cos ( + e^, cos ,T 2 = cos a 2 cos^ + 6 2 ), . . . , cosu^j = cos a n _ 1 cos (s n _ 2 4- e n _i)> a 'n == 8 n-l + n> sin a t tan s 1 = tan (s -f fj), sin or 2 tan s 2 = tan fa + e 2 ), . . . , sino^.jtans^ = tan ( /4 _ 2 4- e M _,). (1G4.6) If we take j r cos a? x , 2 = r sin a\ cos i^ 2 , , r sin ^j sin o? 2 cos C :i , n = rsinaj 1 ... sina' 7? _ 1 cos,r ?i , w+1 we see that ! +I + ."+n+ and we easily verify that (164.7) The 7i- way space of curvature positive unity is then the section of an (?i + l)-way Euclidean space by a sphere of radius unity. 165, Geodesies as circles. We shall now prove that every geodesic is a circle of unit radius in ordinary Euclidean space of three dimensions, but generally two geodesies will not lie in the same Euclidean three-fold. We have for a geodesic sin x r cos s r sin a,, cos (^-.j 4- e y .), sin x r sin s r = sin (s r _ l + e r ), and therefore sin x r cos x r+l = A f cos s r _ 1 + B f sin ts r ^ l = a r cos( r _ 1 + J .) + 6,.sin(8 r _ 1 + r ), (165. 1) where -4 r , B r , a r , &,. are some constants. It follows that sin cc r sin x r+1 cos cc,. +2 = sin a 1 /. (-4,. +1 cos 8 r + B r+l sin s r ) = J. r+1 sin a r cos (8,..! 4- e r ) + J5 y . +1 sin ^..j + e r ), and therefore b r cos a r sin x r sin x r+l cos ^ r4 . 2 fe r+1 cos a r sin j' r cos a? r4 1 = (JB r+1 (f r sina r 6 r -4 r+1 )cosa: l .. (165.2) 238 tt-WAY SPACE OF CONSTANT CURVATURE We thus have a linear relation between the three coordinates /> s/'+l' br+2* By a linear transformation in the ('Ji-H)-way Euclidean space t ... w+1 wo can take it that the first such relation is 1 = ' and that ? +j + ... +* + i = * Proceeding thus with respect to any one geodesic we can take it that the equations which define it are i.1 1 that is, ^ t = , ar a = , ... a? M-1 = t t & It is therefore just a circle in the space given by '' 2 = d+'*!5 + i. 065 - 3 ) and its equation is j{ +^- + 1 1, (165 . 4) with & = 0, & = 0, ... _, = (). 166. Geodesic distance between two points. We shall now find an expression for the geodesic distahce between any two points in the u-way space whose measure of curvature is positive unity. Let the two points whose coordinates are a, ...#,* and y^...y n be denoted by x and y y and cousider the geodesic which joins the two points. Let s, s lf ... ,s /l _ 1 be the arcs which correspond to x, and s', s' 19 ... s'^-i the arcs which correspond to y. We have cos x l cos 2/ t + sin x l sin y L cos 5 t cos ft\ + sin ^ sin y l sin ,_ sin s\ = cos 2 a, cos (s 4- e^ cos (s' -f fj) + sin 2 j cos (s + 6j) cos (' 4- 6j) + sin (s -t- 6 t ) sin (' + l ) t and therefore cos(s' s) = cos x l cos 2/j 4- sin x l sin y t cos (8\ s l ). (1GG . 1) Similarly we see that cos^'j si) = cos cc 2 cos 7/ 2 4- sina? 2 sin y cos (8' 2 sj (166 . 2) (1C6.3) GEODESIC DISTANCE BETWEEN TWO POINTS 239 It follows that, denoting the geodesic distance between the points x and y by (xy), cos (xy) cos x l cos y l + sin x l sin y l cos cc 2 cos y >2 + sin 0,'j sin y l sin # 2 sin y 2 cos # 3 cos y., 4- . . . + sin x l sin ^ sin x^ sin ^ ... sin j? M-1 siii y ll _ 1 cos (x n -y n ). (166.4) This is the formula which is fundamental in the metrical geometry of w-way space of curvature positive unity. 167. Coordinates analogous to polar coordinates. We can now employ a system of coordinates, to express geo- metrically the position of any point in our space, which will be analogous to the use of polar coordinates in ordinary Euclidean space. We take any point in the space as origin, that is, the point from which we are to measure x l , the geodesic distance from the origin. It will be convenient to denote this distance by tan" 1 r, so tliat r = tana: 1 . (167.1) Let us now consider the system of geodesies which pass through this origin. For any one of these geodesies # 2 , ... x fl are fixed, and we may therefore regard *r 2 , ...x n as the co- ordinates which define the geodesic, and thus regard r, a' 2 , ...x n as the polar coordinates of a point in our space. The geodesies through the origin cut the surface r = con- stant iii an (?i l)-way space of positive curvature 1 +r~ 2 . In particular the surface r = infinity (167 . 2) is an (/i l)-way space of curvature positive unity, and the coordinates of any point in this space define a geodesic through the origin. The geodesic distance between two points at smull distances x l and y 1 from the origin is given by cos (^2/1) = cos x l cos 2/! + sin x l sin y l (cos x 2 cos y 2 + sin x 2 siny 2 cos # 3 cos 2/ 3 4- . . .), and therefore t X 2 1/ 2 240 ?-WAY SPACE OF CONSTANT CURVATURE But, if is the angle between the geodesies through these points and the origin, (^2/i) 2 = x l +2/1 - 2 ^i 2/i cos 0- It follows that cos Q = cos x 2 cos // ,, -f sin :c sin //,, cos # a cos y 3 -f . . . , (167.3) that is, the angle between the two geodesies is the geodesic distance between the points where the geodesies intersect the surface r = infinity. The geodesic distance between any two points is therefore the geodesic distance between two points, on a sphere of unit radius, whose polar distances from a point on that sphere are #! and 2/1, and the difference of whose longitudes is the angle which the geodesies through the points cut out on the surface r = infinity. 168. The three-way space of curvature -f 1. We now limit ourselves to the case where n = 3, that is, the three-way space of curvature positive unity. For this space x is the geodesic distance from the origin ; and x (2 and x (} may be taken as the polar coordinates of the point on the two-way surface of positive curvature unity, ./'! = . where the <L geodesic, through the point x, x%, x 3 and the origin, inter- sects the surface. We may without loss of generality suppose that x l lies between and , x. 2 between and TT, and # a between and i 2n. In the surrounding four- way Euclidean space x will then always be positive. Through 'two points in our space one, and only one, geodesic can be drawn, unless the two points lie on the same geodesic through the origin, and are the two points where that geodesic intersects the surface x 1 = Through three points in the space we can in general draw one, and only one, two-w&y locu^s of positive curvature unity. THE THREE-WAY SPACE OF CURVATURE +1 241 We see this by noticing that three points (x l9 ;r 2 , x 3 ), (y l> y 2 , y^), and (z lt 2 , s :j ) determine the plane in the surrounding Euclidean space. The exceptional case would be when the three points lie on the same geodesic. By a linear transformation, in the Euclidean four-way space, we may take the plane to be t = and the locus of the points of intersection with the sphere to be given by There will then be a corresponding set of coordinates o/p .r 2 , r, such that tlie locus is given by x l = in the new *"" u coordinate system. It will be convenient to call any^ two-way Jocus of curvature positive unity a plane, though we should remember that it is only properly a plane in the Euclidean four-fold. Similarly we shall call any geodesic a line. Plane geometry in our space is therefore just spherical trigonometry. 169. The geometry of the space. We may now introduce a different system of coordinates in order to bring out the relationship between the geometry of space of curvature positive unity and that of ordinary Euclidean space. Let x tan n\ sin x 2 cos ,r ; , , y = tan x { sin .r 2 sin ;r 3 , z tan Jc l cos t r 2 . (169 . l) In this system of coordinates the geodesic distance between two points (x, y, z} and (,', y', z') will be cos" 1 ^-- t -i~^ yri ' " (169 . 2) where r 2 = x 2 -f y 2 + z*. The square of the element of length will be given by dtp = (1 -f r 2 )" 1 (dx* + dy 2 * 4- dz* ( -f r-)~ l r 2 dr-) ; (169 . 3) but in this geometry, as ir) EucMcan geometry, having the 242 7fc-WAY SPACE OF CONSTANT CURVATURE expression for the actual distance between any two points, we do not need to make so much uso of the expression for the element of length. The equation of any plane is \x + p.y + vz + 8 0. Now a plane, we know, is a two-way surface of curvature positive unity. Let x lt y v z l be the coordinates of its centre, that is, the point at a geodesic distance - from every point u of it. We then have sx l + yy { + zz +1=0, (169.4) and therefore Bx l = X, By l = //, 8z l = v. The angle between two planes is, as in spherical trigonometry, the supplement of the angle that is, the geodesic distance between their centres. The cosine of the angle between the two planes ,1 r 12 15i , v ,,, is therefore l - 2 ~~ f ^^\^ l 2 ~ l - - (1 GO 5) (Xf +f*i+vl + Sir(W+Ltl+i> 2 > + 8rf The equation of a plane, given in terms of the coordinates of its centre, is ^ + yy v + ^ + 1 = 0. (1 G9 . 4) The condition that the plane passes through the origin, that is, the point where x, y> and z are each zero, is that its centre should lie on the plane whose centre is the origin. The equation of a line is given aw the intersection of two planes tCXl + yy l + ^+1 = 0, ^2 + 2/2/2 + ^2+ 1 0. In connexion with this line we consider the line joining the points (x lt y^ X ) and (u? 2 , # 2 , z >2 ). The plane whose centre is A may be called the polar plane of A. We see that if J5*lies on the polar plane of A, then A lies on the polar plane of .& THE GEOMETRY OF THE SPACE 243 We now see that if (x l9 y^ X ) and (x, 2 , y 25 z 2 ) are any two points on a line, then every other point on the line is given by Z = 1* 9 (1G9.6) J where p : q is an arbitrary parameter. The line given as the intersection of the planes ^+1 = 0, stands therefore to the line joining (x ly y ly z^) and (# 2 , y^ 2 ) in the relationship, that the distance between any point on the one line and any point on the other line is - . The lines Li which are in this relationship will be called polar lines. We now wish to consider two lines, viz. the line given by C3 2 +l = 0, and the line given by ^4 + 2/2/4 + S3* +1 = 0. If these lines intersect, the four points ('''l i 2/1 . C l) 5 ('<'*> ?/2^2)> (^3 2/3 . -a)> (^ 4 #4> -4) lie on a plane, and we thus see that if two lines intersect their polar lines also intersect, and the plane on which they lie is the polar plane of the point of intersection. 170. Formulae for lines in the space, and an invariant. Just as in Euclidean geometry, a line has six coordinates. We define these coordinates I = # 2 - iH t , m = 7/ a - y l , n = z, -z v X = y^ - y^ , The six coordinates are those of the line joining the points (x l , I/,, , j) and (ic 2 , ?/ 2 , ^ 2 )> an< ^ ^ e y are connected by the relation ZA + w/*-Hii'= 0. (170.2) We easily see that if Z, m, w, A, /i, V are the coordinates of a line, the coordinates of itfj polar line are A, /z, i/, ^, ?>i, n. 244 71- WAY SPACE OF CONSTANT CUKVATURE If (12) denotes the geodesic distance between the points COS (12)=- y^y, ' J* and therefore sin' (12)= m + tt + + j* + . ( } ' (l+^i)(l+^) ^ ^ Consider the expression IV + w/m' 4- < ?m / + XX" + / + i/^ _ /* 2 + A- + / 2 + i^*(^^ where (Z, m, -H, A, /^, i/) and (/', m', /i', A', /, i/') are the co- ordinates of the lines which respectively join the points 1 and 2, and the points 3 and 4. It is easily verified that the numerator of the expression is and the denominator is ^l 1 + 6 'i) (1 + *y I 1 + 6 '.;) (I + ^4) ^' m (12) sin (34). The expression is therefore equal to cos (13) cos (24) -cos (14) cos (23) sin (12) sin (34) ' (170.4) and this is clearly an invariant. Suppose now that the points 1 and 3 coincide. The ex- pression becomes cos(24) - cos (14) cos (12) sin "(12) sin (14) ' and we<see that this is the cosine of the angle between the lines 12 and 14. Suppose next that the 6 line 13 is perpendicular to the lines 12 and 34. FORMULAE FOR LINES IN THE SPACE, AND AN INVARIANT 245 Clearly, from the formula cos (24) = cos (14) cos (12) (170.6) when the lines 12 and 14 are perpendicular, the line 13 will be the shortest distance between the lines 12 and 34. The planes 132 and 134 will be the planes through the shortest distance and the lines 12 and 34. We may, to interpret the expression W + in m' + 'tin' + XX' + fifi + i/i/ since we have seen that it is an invariant, take the points 1, 2, 3, 4 to be 0,0.0; <r 2 , 0, 0; 0, 0, s., ; 4 ,?/ 4 ,c a ; it now becomes ^ = The equations of the planes 132 and 134 become respectively The angle between these planes is cos" 1 JL_ The shortest distance between the lines is 1 cos~ and therefore the invariant expression is equal to the product of the cosine of the shortest distance between the lines into the cosine of the angle between the two planes drawn through the shortest distance and the two given lines. The invariant vanishes if the lines are polar lines. It also vanishes if the planes through the shortest distance and the two lines are perpendicular. If the lines are not polar lines and if the invariant vanishes, we see that the polar line of 12 intersects 34 and the polar line of 34 intersects 12. 246 ft-WAY SPACE OF CONSTANT CURVATURE 171. Volume in the space. The expression for the element of volume in a space of three dimensions and with the measure of curvature positive and equal to unity is sin 2 x l sin .r^ dx l dx. 2 dx. 3 , (171.1) returning to the original notation of 1G8. The volume enclosed by an area of any plane that is, a two-way surface of curvature positive unity and the lines joining the origin to the perimeter of the area is los ojj) sin # 2 (L'2 dx } , (171.2) where x l is the geodesic distance from the origin to a point within the perimeter. If the plane is at a geodesic distance p from the origin we can use the equation tan p = tan x l cos x. 2 , and express the above integral in the form P X 7 i \\P~~ ^ an P x i c t x i ' ( '''> > (171.3) J where x 1 is now the geodesic distance to a point on the peri- meter from the oiigin. If we take r to be the geodesic distance of a point on the perimeter from the foot of the perpendicular, and take x [} to bo the corresponding longitude 6 in the plane, the above formula becomes - p / tan p cos p cos r cos" 1 (cos p cos r) \ A 2 ( /> =-^ 7- )<10- (171 . 4) J v VI cos p cos" >' / If the foot of the perpendicular lies within the area, this formula gives us for the volume the expression fsin p cos r cos" 1 (cos^ cos r) fi pTT - ~~ j= -~=^= (6C7, (171 . 5) J V 1 ~COS Z ^> COS- ?' where the integral is to be taken round the perimeter. We notice that in space of curvature positive unity when s, the variable in the equation of a geodesic, increases by 2 TT, then 8 1? s 2 , ... also increase by 2 TT, and therefore the coordi- nates #, , . . . x n all increase t>y 2?r. cWe thus, in proceeding along VOLUME IN THE SPACE 247 a geodesic, come back to the point we started from. We cannot have any two points at a greater distance from one another than TT. 172. An ti-w&y space of constant curvature as a section of an extended Einstein space. We now wish to consider the 'ii-way space of constant curvature as a section of an (/6 + l)-way surrounding space. We take the (71+ l)-way space ground form to be pdut + bftdXidx!., I = 1 ...>*, (172.1) where b ik = a ije when u 0. We have ( 146) (r- A-) 6 - Extending the definition of an Einstein space, wo shall now say that a space is an Einstein space if bM(rkih) = cb lh , (172.3) where c is a constant. We have %. (172.4) If the surrounding (u+l)-way space is to be Einstein space according to the new definition,* we must have [Called iii 18 311 extended Einstein space. \ 248 W-WAY SPACE OF CONSTANT CURVATURE ca ri = J rt + * ft (/2 rt J2 4fe -/ That is, if Ff = u kt n tl (172.5) where /2, 7 , = /2 H = a,, x F* = </, <A F*, we must have c = $-* (\ ((f>)-a ri ^ fl, ; ) - F* F^ , (1 72 . G) VJ.,= F M . (172.7) Now ^^^ and therefore Multiplying by ct'^ and summing, ref = A V , + v j ; v + <f>- 1 0J 1 - -- F'; (172 . 8) We also have That is, we have ~ We may replace the equation iV=**<* by ^ + F 2 - F F^ = (TI - 1) c. A SECTION OF EXTENDED EINSTEIN SPACE 249 The equations therefore become A + V 2 -V*V = (n-l)c, (172.9) F^ = ^ ( (172.10) = _20 ft y;. (172.12) We may easily verify that the results which we have proved for the case c = still hold in this more general Einstein space. The special conditions that the coefficients b^ and the function may be independent of u become A = (M-l)c, H + = c0, that is, 0{Afy*} + 0.AM 0tfA/n A 2 = 0. (172.13) Now let us assume that the ?i-way space is of constant curvature K. We have {\ttfi} = (l-n)A r ra/z. If we choose /i so that //& = -C, the conditions that the surrounding Einstein space may satisfy the required conditions become ^ + Ka^ = 0, Az(<f))+Ku(/) = 0. The second condition is a consequence of the first set, and we see that all that we need is that the system 0.A/i + A r a A Ai0 = (172.14) may be complete. We know it is, and thus we may take </> = cos# 1} and the space o'iven by cfe 2 = cos 2 ^ du z + R- (dx \ + sin 2 ^ dx * + . . . ) (172.15) will be an Einstein space of the kind required. If the space is of constant negative curvature we should see that regarded as a locus in (n 4-1)- way Euclidean space it would be an imaginary section. The expression for the geodesic distance in space of negative curvature unity is given by cosh s + sinh x 1 sinh y l + cosh x l cosh 2/^sinh # 2 sinh y 2 + , . . f sinh x l . . . sinh x n _ 2 sinh y l . . . sinh y n _ 2 cosh x n ^^ cosh l' n - l ... cosh x n _pco*>h[y l ... cosh y n - l cosh (% n --y n )' CHAPTER XIV vi-WAY SPACE AS A LOCUS IN (n+i)-WAY SPACE 173. A space by which any u-way space may be sur- rounded. We now consider again the ground form of an (>b+ l)-way space du* + b ik dx i (lx j . which, when we put u = 0, becomes a^dXfdx^. We have, by the formulae of 146, {Mi},, = {ikh: a - {ik-] b = n {h -, {i-k\ b = - = * + 2/2, . d a lf> \V f e shall prove that we may surround any 91-way space with a space for which (^')ft = 0; (r-i-) h = 0. (173.1) ** = -% then A/, //>fc = (* / n H - {rw, n ti - { ikt } n lr ) \rkt} -n t , {Hot}. A PARTICULAR SURROUNDING SPACE 251 Now we have seen that and therefore that is, A*.* = x * (^A-fc-^A-'Vfc) = 0. (173,2) Thus we see that the relations fl ri .^ = -^iVi- persist when U. and lO row in accordance with the laws +a x "/2,. x /2^=0, (173.3) 7^ + 2^=0. (173.4) We can therefore surround the 7i-way space with a space for which ( r ]>i.) b _ o ; (r-i-) b = 0. 174. Curvature properties of this surrounding space. We will now consider what properties such a surrounding space would have as regards curvature. Consider the ground form of the surrounding space, which we denote by the suffix ft, Let B 7; = i;x? l + ... + v be two vectors of lengths j | and | rj \ inclined at an angle & which lie in the tangential (^+l)-fold and therefore in Euclidean space. Let = i 0i + ...+ 5 be two corresponding vectors of lengths | | and | TJ \ inclined at an angle 6 aad lying in^he tangential n-fold. 252 71- WAY SPACE AS A LOCUS IN (+l)-\VAY SPACE Wo have I I* = ''*, 4 | | 2 ! r, | sin* = (V-|V) (V-V) (-% V- The measure of curvature 7f a , according to Riemann, which corresponds to the orientation given by the vectors and ?;, satisfies the equation 4 sin* 6 | | 2 h | 2 A' a - ( V- V) (V - Now consider the vectors , 77 when u = 0. i;| cos*) = 0, We therefore have snce But ( aud therefore n pk - n ilt n M ). (174.1) Here K b is the Riemann curvature in the (ra-f l)-way space corresponding to the orientation of the vectors + z and rj 4- 772; ; and iT a is the Riemann curvature in the '/i-way space corresponding to the vectors and 77. 175. We may express the result in yet another form. Consider the ground form cfl^dx^dx^y where c is a constant introduced to keep the dimensions right, and let a vector ( be defined by the equations /.A.-f r/2-/ = 0. The vector will then trace out in some Euclidean r-fold an 7i- way space vc l ...x n . * c CURVATURE IN THE PARTICULAR SURROUNDING SPACE 253 In this space let us consider two vectors and 77 of lengths | | and 77 | inclined at an angle 0', where We have 4sin 2 0' | 2 1 77 1 2 = *('y-V)(W and therefore 2 !r;i 2 + >) 2 !| 2 -277!!|77 cos0) = 0. (175.1) We see that the curvature of this (71+!)- way space which surrounds the given ?i-way space depends, then, on the know- ledge of the ground form fl^dx^dx^ with the property that 176. A condition that the surrounding space may be Euclidean. We now ask whether the surrounding space can be Euclidean ? If it is Euclidean we must have . 0. We have seen in 173 that, if when u the equations H ri . ll = fl rk . i ^ (176.1) hold, they will persist for any value of u whilst n ik and a,-/, grow in accordance with the laws \^ (170.2) = 0. (176.3) 254 7/,-WAY SPACE AS A LOCUS IN (u + l)-WAY SPACE We shall now prove that if these equations hold, then, if (rkih)+n rh n ik -n ri n hk - o (ire . 4) holds when u = 0, it also will persist when u has any value. The expression ~ (rkih) + {rtih} n ]d - {Idih} fl fl is a tensor component : when we refer to the geodesic coordi- nates of any given point we shall find that it vanishes at that given point and therefore vanishes identically. So referred, (rkih) = - ( (r ik) ( ii i Now = Sl fi . k + { rkt } n u + { ild} n, t , and therefore ~ (rkih) = A ( [ikt}il rt - { irt] n M ) = f2 rt [ktlh] -n l:l (rtih], (176 . 5) which proves the required formula. Again fX a^ (kphi) CONDITIONS FOR A EUCLIDEAN SURROUNDING SPACE 255 It follows that ^ ((rkih) + n^n^-n,^) = o ; that is, the equations Sl ik '-fl ti fl Ilk = 0, (176 . 6) if true when -u = 0, will always be true. The condition that an ')i-way space may be contained in a Euclidean (H + l)-\vay space is that the equations n ri . jf = n tjf . i9 (17G.7) may be consistent. 177. Procedure for applying the condition when n > 2. There is now an essential distinction between the case n 2 and the case n > 2. A two-way space is always contained in a Euclidean space of three dimensions, and we have considered the problems associated with this case. If 71 > 2 we can uniquely determine the functions fl ik in terms of the four-index symbols of ChrisloiFel, by aid of the equations alone. If n > 3 we even have relations between the four- index symbols from the consistency of these equations. It is a problem of algebra merely to determine the functions /J^, and the functions so determined arc tensor components. If the surrounding space is to be Euclidean, the functions so determined must satisfy the equations f2 t ^.j. = f2 r j { .^ We can therefore, when we are given the ground form ^^t/.r^fo/., determine, by algebraic work merely, whether the space to which the ground form refers is or is not contained within a Euclidean (91+!)- way space. The actual work would, however, be laborious. 178. The n-way space as a surface in the Euclidean space when this exists. Suppose, ^iow, that we are given the ground form a^dx^lxj^ and that we have found that the space 256 n-WAY SPACE AS A LOCUS IN (tt+l)-WAY SPACE to which it refers is contained in a Euclidean (?i+l)-way space and have calculated the functions fl ik : we may ask, what is the surface in Euclidean space which is the given n- way space? Let z bo the vector in the Euclidean (n-fl)-way space which traces out the given ?i-way space. We know from our earlier work that z. ik is normal to each element of the space drawn through the extremity of z. Now there is only one such vector in the Euclidean (?i+ l)-fold. Let A be the unit vector which is normal to the surface. Then -to = ./, A, (178.1) where w^ is some scalar. We have ~ ., _ w .\. 4-?/> . , \ " ~ and therefore, since kl-lk - \tpik\ (p- \qtik\ (J), where p is an upper integer and q a lower integer, we have - [Ttik\ Z t = W f i\k-Wrk*i + (Wri-k-- w rk*i)*- ( l 7S ' 2 > Multiplying by A, and taking the scalar product, and noting that X\ p = 0, we have v , f . fc = Wrk ^ We also have - = (rkhi), and therefore w rh w ki~~~ w ri' w kh ~ It follows that w ik = /2 {/ ., (178 . 3) and we have z. ik /2 l7 ,A. (178. 4) We also have [rtik\ z t = /2 / . 7i .A i -/2 /{ A / ., (178 . 5) so that when we know A we can find z by quadrature and thus determine the surface save for a translation. 179. We have now to show how to determine A. Xz p = 0, and therefore ^g^ + Xz. 0. It follows that XjZfr = t\^ = fl ik . ^~ ^- t W-WAY SURFACE IN THE (tt + l)-WAY EUCLIDEAN SPACE 257 From the equation {rtik} z t = fl fl6 \i-Sl ri {rtik} fl tp = n rlc \i\ p - fl and therefore tt (rryi/c) /2, p = n, rk \j\ p -{2 H \ k \ l) that LS, c*'tf/2, p (fi ri f2 qk -fi rk f2 qi ) = ^x^-^x^, or /2,,. fe (AjA + a'S-fyp/fy) = ^(X + c^/2/2). (179 . 2) Unless, then, the coefficients of f2 rk and /2^ are zero, we must have n rk n gi = n ri n sk , which would mean that (rski) = and that the 76-way space was Euclidean, a case we need not consider. We conclude that \i\ k + aP ( in, pi n qk =0. (179 . 3) We thus know the ground form of the surface traced out by the unit vector X. Let \. ik = ^~ -(ikt>'^> (179.4) ** *xt*x k <>^ t where {ikt\* is formed witli reference to this ground form. We have AX; = 0, and therefore XX.^-hX^X^. 0. Now \. ik is parallel to the normal to the surface traced out by X, and therefoi*e, as X is a unit vector, is parallel to X. it follows that A.^-ha^/i^^X = 0. (179 . 5) We thus have the equations which determine X. These equations may be written A-<* + a '<*A = > (179.6) where a' ik denotes a coefficient in the ground form of X. As this ground form is that of a space of constant positive curvature we see that the system is c complete '. It follows that we can allow X t . . . X n to take any initial values and thus we can determine X save as to a ' movement ' in Euclidean space. 258 ft- WAY SPACE AS A LOCUS IN (n + 1>WAY SPACE We have considered three ground forms ; these maybe written dzdz = dzd\ = d\d\ = We saw (147.4) that the lines of principal curvature were given by the equations (a pq -Rfl pq } dx q = 0, that is now, by z p (dz Rd\) = 0, and as we also have X (dz RdX) = 0, ^ ^/ we conclude that dz = Rd\ (179.7) is the equation of the line of curvature corresponding to the principal radius of curvature It. INDEX (The references are to pages) Applicable surfaces, defined, 27, determination of, 67. examples of, 26, 28, 31. Associated vector spaces, 1 7. Asymptotic lines, reference to, 77, 79. on minimal surfaces, 153. Beltrami's differential parameters, 16, 17, 56, 80. are invariants, 17. Bertrand curves, 116. Bianchi, 132, 171. Bonnet's theorems on ruled sur- faces, 131, 139. Central surface of a congruence, 97. Christoffel's three-index symbols, 6, 7, 56. four-index symbols (tensor components), 19, 20, 21. Codazzi's fundamental equations, 64, 65, 67, 77, 132, 135. moving axes equations, 113, 125. Complex functions of position, 59. Confbrmal representation, 155, 157. on triangular area, 158, &c. on a given polygon, 168. Congruences of straight lines, 93, &c. of normals, 104, 107. Conjugate harmonic functions, 60. lines with equal invariants, 83, 92. Constant Riemauu curvature, 232. Contracted notation, 1, 4, 6, 8, 12, 62. Curvature and torsion of a curve, 110, 113, 180. Curvature, lines of, 66, 68, 76, 105, 153, 179. measure of, 39, 49, 64, 69, 181. principal ladii of, 64, 230. principal (rc-way), 208, 209, 219, Curved space, the idea of, 2. Darboux, 40, 56, 113, 171. Derivatives of a ik , &c., 8. Developable surfaces, 30, 68. 1 Erection cosines, extended notion of, 5, 76. Dummy suffixes, 6. Einstein space, 23, 212. extended, 24, 247. four- way stationary, 228. (n+ l)-way stationary, 220, &c. (n+ l)-way surroundings- way, 207, 212. Elliptic coordinates, 31. Enneper's theorem, 125. Equivalent ground forms, 32. Euclidean coordinates at a point, 18, 21, 46, 190. surrounding space, 2, 207, 253, 255, Focal coordinates, 55, 72. lines non-existent, 100. glanes and points, 95. surface of a congruence, 98, 105. 260 INDEX Four-way stationary Einstein space, 228. Fundamental and associate ten- sors, 14. magnitudes, the two-way Q n , 12 12 , M> 63, 65. magnitudes, the w-way 12^, 207, 212. Galilean coordinates at a point, 193. Gauss, 65, 194, 198. Geodesies, 42, 43, 54, 237. in ra-way space, 188, 199, 203. in w-way space of curvature + 1, 236. Geodesic curvature, 45, 51, 53, 119, 208. distance (w-way), 238. equations integrable when K is constant, 54. orthogonal trajectories, 44. polar coordinates, 192. torsion, 120. Ground form, with K constant, 34, 36. with A(#) =0, 37. with A (K) and A 2 (/r) functions of K, 37, 38. general two-way, 39. Hamiltoniau equation, 96. Harmonic functions, 60. Helicoid, 117. Henneberg's surface, 151. Hypergcometric series, 158. Integration of geodesic equations with constant /C 54. Invariant K, 32, 56, 64. Invariants, systems of, 23, 24. Isotropic congruences, 102. the ruled surface of, 141. Laplace, 83. Lie, 108. , Limiting points of ray of con- gruence, 96. Lines in three-way space of curva- ture + 1, 243. Liouville, 58. surfaces, 228. Malus's theorem, 101. Measure of curvature (see Curva- ture, and Invariant K). Riemann's oriented hi ^-way space, 194, 198. Mercator's projection, 61. Mourner's theorem, 120. Minimal surfaces, 70, 92, 103, 143, &c., 155. associate and adjoint, 153. double, 151. Moving axes, 125. n-way principal curvatures : orthogonal directions of, 208, 229. sum of products of, 209. n- way space : in a higher Euclidean space, 4, 17. in (w-f l)-way space, 206, 250. in (?i-f l)-way Einstein space, 212, 219. 111(71+ l)-way Euclidean space, 253, &c. of constant curvature for all orientations, 232, 247. of zei o Kiemanu curvature, 231. Normal vector, 17, 63, (>5, 78. Null curves, 147. Null lines, 58, 145. Orthogonal burfaces, 127, 172, c. intersect in lines of curvature, 179. special classes of, 185, 187. Parallel curves, 51. Plateau, the problem of. 155, 168. Polar coordinates in n-way space of curvature + 1, 239. geodesic coordinates, 47. INDEX 261 Principal planes of a lay, 97. surfaces of a congruence, 97. Pseudosphere, 27, 75. Pseudo-spherical trigonometry, 2 7. Quaternion, 174. notation, 62. Rank of a tensor, 15. Reduction of a two-way form : with constant K, 34, &c. with &(K) = 0, 36. with A(]r) and A 2 (#) functions of J{, 37. in general, 38. Refraction of a congruence, 1 00. Regular solids, 167. Ribaucour, 93, 101, 102. Riccati's equation utilized, 39, 41, 108, 109, 115, &c. Riemann, 171, 194. Rotation functions, 40, 111, 178. vector, 87. Ruled surfaces, 28, 129, &c. anharmonic properties on, 138, 139. applicability on, 133, 135. ground form for. 130. line of stiiction on, 137. which cut at one angle along a generator, 140, 142. Scalar product, notation for, 4, 62. Schwartz, 171. Second derivatives in transforma- tion theory, 1 1. Self-conjugate null curves, 148, 151. -T- j/iQiyl i' P'l'i'n' 1 1 ii 1 1 and torsion formulae, 110, 114. Space of curvature -f 1, 237, 240. of curvature zero, 231. Spherical image, 66, 73. trigonometry, 27. Surfaces of revolution, applica- bility on, 26, 107. orthogonal to geodesies, 199, 203. with plane lines of curvature, 122. Tangential equations, 69. Tangential Euclidean n-fold, 191. Tensor calculus, rules, &c. of, 14. components, defined, 9. generated from others, 21. derivatives, 12, 13, 16. Tensors, fundamental and asso- ciate, 14. Three-way space of curvature + 1 , 240, &c. Tractrix, 28. Translation surfaces, 150. Twelve associated surfaces, 89, 9 1 . Upper and lower integers, 7. Vanishing invariants of Einstein space, 23. Vector, the rotated qOCq~ l , 174. of triply orthogonal svstem, 178. Vectors in a Euclidean space, 4. ets of orthogonal, 175. Vector-products, notation for, 62. Volume in three-way space of curvature -f 1, 246. Voss, 208. W congruences, 103, 106, 107. W surfaces, 71, 72, 106. -Woki^rass, 171. Weingarten,""71 Zero Riemann curvature, space of, 231. .